[Edit. I originally answered the wrong question, which in my head was about whether (half + plural noun) takes a singular or plural verb. While that does bear upon the current question, it's only half the problem. Here's my new answer - if it helps!]
Over half our board and staff is people of color.
Over half our board and staff are people of color.
The first question which need to be addressed, is whether plural Noun Phrases with the determiner half cause singular or plural verb agreement. We can investigate here using the Noun Phrase half the people. If as some comments here have suggested, half causes singular verb agreement, then the Original Poster's question will have been resolved. If it causes plural verb agreement, then we will still need to consider whether the co-ordination "board and staff" should be treated as a singular or plural subject.
With regard to the first question, some hard talking empirical facts might be in order:
A Google search for:
"over half the people was"
... with the phrase in quotation marks as in the example, returned a staggering three hits! Two of these were along the lines of "the average age of over half the people was 2.14..." So that reduces it to one valid hit of the type with which we're concerned here.
On the other hand the same search for:
"over half the people were"
... gives a return of 245, 000 results. Now on the basis that grammar is what people actually do when they speak, this means that Over half the people were is 245,000,000 % more likely to be grammatical than Over half the people is.
If further evidence is required, the number of instances of Over half the people were from Google books is 2,100. The number of instances of Over half the people was, surprisingly, is 1. However, this single example on close inspection turns out to be in the form of the following:
The standard of living for over half the people was beneath what would have been considered enough for an animal in France or England.
Here the subject of was is the non-countable Noun Phrase standard of living. This reassuringly puts the number of published examples of Over half the people was - specifically of the type we're after - at zero.
So purely on an empirical basis, we can discard any notion of Over half the people grammatically occurring with a singular verb form! Full stop.
This seems to clear up the issue here of whether [half + plural noun phrase] should take singular or plural verb agreement. However, we now have the more pressing concern of whether [the board and staff] as a co-ordination should actually be considered singular or plural.
Both board and staff are collective nouns. Despite some prescriptivists trying to insist that singularly inflected collective nouns should take singular verbs - this is particularly popular with the noun staff -, this so-called grammar rule has never been true. It has always been the case that sentences with collective nouns like government or team as subject can take either plural or singular verb forms. We will commonly find sentences of both types in both formal and informal speech and writing. Here are examples from formal writing:
A building in England was loaned to the project and the staff were drawn
from the Hampstead Nursery run by Anna Freud and the English reception camp
(for refugees) to which the children had been sent.
staff was not present during coffee time.
However, the board were particularly interested, correctly so, with regards to the balance ... .
[T]he board was faced
with the stark reality of having to buy its way back into the business it had once owned.
This shows immediately that plural verb agreement should be permissible with our co-ordinated subject. Our co-ordinated subject is after all a co-ordination of collective nouns. However, we still need to settle the viability of singular verb agreement with our Noun Phrase co-ordination.
It should be noticed that both board and staff are countable. Compare for example:
- We have a staff of fifty
- *We have staff of fifty
The unacceptability of the second is due to the fact that singular countable nouns must be preceded by a suitable determiner. Staff in the second example is missing a. This requirement does not hold with non-count nouns:
Co-ordinations of countable nouns in English usually occur with plural verb agreement:
- *The husband and wife goes to church every week.
- The husband and wife go to church every week.
There are exceptions to this. For example when co-ordinations are seen as one joint amount or entity:
- bread and butter is my favourite dish.
This does not appear to be the case here, where board and staff are considered discrete - or as CaGEL, (Huddleston & Pullum, 2002; 1283) would put it, distributively. Notice also in the exceptional example above that the complement of BE is also singular. This is not the case in the current example. This would seem to rule out our board and staff from being singular. Again we could back this up with some empirical evidence from Google Books:
the board and staff was
This receives six hits, in none of which the board and staff is the subject of the verb.
the board and staff were
This search, in contrast, receives twenty-three hits, in twelve of which board and staff is subject - with plural verb agreement. This is as we would expect. A co-ordination of countable nouns will generally require plural verb agreement.
In answer to the main question then, only the following should be deemed grammatical:
Over half our board and staff are people of color.
This still leaves the question of the following examples:
Over half our board [are/is] people of color.
Over half the stadium [are/is] people of color.
We have seen that half does not affect verb agreement with the subject noun phrase. We have also seen that group nouns can have either singular or plural verb agreement. This should make either choice possible. However, there is one further issue to consider. When we have collective nouns the predicate can influence whether a singular or plural is more felicitous. Here the predicate includes the Noun people. This is very different from if it had been an adjective:
The above seems fine. The example below is less acceptable:
- #The board has all received their retirement packages.
In the latter sentence, the predicate necessitates considering the board as a number of individuals. Whilst it might be argued that the second sentence is not felicitous, some would argue that it is still grammatical. (However, it still not felicitous!)
The fist sentence has no such problem. It is a matter of individual judgement, but the predicate people, in conjunction with the word board seems to favour the group of individuals reading more strongly than the stadium example. In either case there is no hard and fast rule for sentences with a singular collective noun as subject.
It's only my personal opinion, but prescriptivism can get you into deep trouble. Prescriptivists usually have to depend on their grammar rules not their ears. The fact that some of the Original Poster's colleagues favoured Over half the board and staff is ( - a complete error), is almost definitely due to hyper-correction. This correction is in relation to the prescriptivist rule that singularly inflected nouns must take singular verb forms; particularly the influence of this idea in relation to the noun staff. If they'd jettisoned this unnecessary rule in the first place, then awkward sentences like the one in the title here would be avoided. I would therefore plump for a plural verb form in each case. But, as I said, that bit is just my personal opinion.