In the 1950s we were strongly discouraged from placing prepositions at the end of sentences, and also from using split infinitives. Is this considered important now?
It depends who is doing the considering. Almost all academic linguists recognise that these "rules" have no historical authority but were arbitrarily invented a couple of hundred years ago. Most modern style-guides have abandoned the one about prepositions, and are prepared to allow split infinitives if the alternative would be awkward or ambiguous.
But you will certainly find curmudgeons about who insist on these "rules"; and of course if any such curmudgeon encounters your prose which doesn't follow their cherished rules, they will judge you as inferior.
English has pre-positions ("We looked at the books") and post-positions (The books we looked at"). Why forbid the second ones ?
As for split infinitive, "Writers should learn to not split infinitives".
It depends on how formal you want to be. If you're writing a speech for a large educated group - let's say college professors at a conference - then you'd follow both rules.
The infinitive thing is a carryover from Latin, in which the infinitive is a single word ("amare", to love). In Latin, you couldn't split it if you wanted to. When things started to be translated from Latin, they noticed that you always wrote "to be" for "esse" (for eample), and somebody decided that it would be bad form to write put anything between the "to" and the verb.
As it's a rule that no longer makes sense, we ignore it.
As for the ending preposition, Winston Churchill was one of the finest writers in modern England. When someone tried to correct him - ending a sentence with a preposition - he replied
"This is the sort of thing up with which I will not put."
(Versions of that story vary.)
Again, I think it comes from Latin, where the preposition is always in front of the phrase.
English is, however, a living language. That's why we have so many words - we borrow as needed from other languages.
Split infinitives and terminal prepositions have a tendency to add complexity and ambiguity to a sentence, or help reduce them.
Your current attitude should be - does their use in a particular situation increase or decrease comprehension?
For example I encountered this rather confusing advertising phrase (for a telecom product) in Singapore many years ago.
Which line would you rather be at the end of?
It could be rewritten as
At which end of a line would you rather be?
Perhaps, to emphasize differentiation of one product line from another
At the end of which line would you rather be?
However, this phrase with a terminal preposition,
Whom do you think I should be working for?
is more comprehensible than one without,
For whom do you think I should be working?
Whilst (does anyone still use the word "whilst"?),
For Whom the Bells Toll
sounds much better than
Whom the Bells Toll For
These sound so much more grammatically refreshing,
Would you like me to set your equipment up before your demo tomorrow.
Remember to pick the groceries up from the shop this evening.
But, these sound more comprehensible and get the job done.
Would you like me to set up your equipment before your demo tomorrow.
Remember to pick up the groceries from the shop this evening.
Yes, they matter. Sometimes. And sometimes they don't.
The anti-prescriptivists say that such rules are arbitrary, nothing to with English(!), and should be ignored. While doing so they obey a whole load of other rules, so this position doesn't stand up logically: it is cherry-picking which rules are prescriptive and which are not.
The prescriptivists say that all rules should be obeyed regardless, and failure to do so is dumbing down the language, a failure of education, etc. They usually believe that the Golden Age of Grammar was when they were in school. This too doesn't stand up logically as they have to prove that their set of rules is correct and, for example, 17th century rules are not.
Somebody, somewhere, said something like "rules are for the adherence of the ignorant and the guidance of the wise". Maybe that should be applied here.
The 'arbitrary' rules that 'have nothing to do with English' were codified by Englishman who were very well educated and knew more about the language than those who currently whine about them. They are there for a reason, and to arbitrarily dismiss them makes no sense. On the other hand, there is no reason why those who understand the purpose of the rules should not break them.
"To boldly go", for example, is harmless enough and improves the prosody, at least to my ear. "To go boldly" as prescriptivists would insist is comparatively clunky, and as the purpose of not splitting infinitives is to keep the to- and its verb together (not a bad idea) splitting with a single short word doesn't really violate that.
Compare with "to boldly with phasers set to stun, transporter at the ready, and Bones hanging around with his tricorder, go" is almost unintelligible but presumably okay to those who say splitting infinitives is awesome.
In short, the rules themselves are not wrong. The issue is people who either apply them religiously, or on the other extreme think they should be ignored completely.