On the other hand, this source states that:
In British usage, we do not use an apostrophe in pluralizing dates:
This research was carried out in the 1970s.
American usage, however, does put an apostrophe here:
(A) This research was carried out in the 1970’s.
You should not adopt this practice unless you are specifically writing for an American audience.
This is contradicted by a forum post stating that this is simply incorrect.
Now I’m not sure anymore. Considering the sources I’m pretty sure 1960s is correct, but is there any situation, in any recognised English orthography, in which the spelling 1960’s is also correct in referring to the decade (as opposed to a property of the year 1960)?
I should also point out that if the apostrophe is used, it can’t represent “possession/association” unless followed by another noun. You can say I was last year’s winner, and (just about) I was 2012’s winner, but that’s a completely different usage. In OP’s context, the 60’s can only be a noun meaning “that decade”.
Besides which, if you really wanted to use the possessive with the sense “of that decade”, it would have to be the 60s’ singer Pat Boone, for example. But nobody would actually do that – we just assume that regardless of whether it includes an apostrophe, the 60s there is a “noun” used as an adjective.