To my intense embarassment I have to say that this accepted answer is not correct for the sentence “The prevalence of both...”. The subject is prevalence, singular, and therefore the verb must be the singular, ‘was’. I offer my profound apologies.
Answer to a different question
If the sentence had been “Both diabetes and obesity increased with age...” the verb would be the plural, ‘were’.
The ‘and’ in ‘Both X and Y’ makes it clear that ‘both’ has a plural subject and requires a plural verb in normal usage.
I would not have thought it necessary to provide citations to support my statement that ‘both’ requires a plural verb with a plural subject. This seems uncontentious as it is not even treated under the entries for ‘both’ in Gower (‘The Complete plain words’ and ‘English Modern Usage’) or by Fowler and Fowler in ‘The King’s English’). However, as I have been asked for citations, here they are:
“And they were both naked, the man and his wife, and were not
Genesis 2:25 (KJV)
“The Douglas and the Hotspur both together Are confident against the
world in arms.”
Shakespeare, Henry IV, Part I [V, 1]
For both were faiths and both have gone.
Mathew Arnold “The Grand Chartreuse”
That said, I must admit that Shakespeare is (as ever) not consistent. Thus:
Nay, come, I pray you, sir, give me the chain:
Both wind and tide stays for this gentleman,
Shakespeare, Comedy of Errors [IV, 1]
But this strikes one immediately as archaic.