These days they only hang things which are unrelated to Tanabata.
These days, on Tanabata, they only hang unrelated things.
First off, both 「～～に関係ある/ない」 and 「～～と関係ある/ない」 are correct and used equally often. To say there is a difference in nuance would be nitpicking. Thus, in the sentence above, the 「に」 can be replaced by a 「と」 without changing its meaning the least bit.
Exceptions: When talking about having a man-woman relationship, only 「(person)と関係がある」 can be used.
Moving on to your translations..
Your first translation is good though it is a free TL. The original does not say "these days" or "they". The subject of the original sentence is 「もン」, which refers to the decorations themselves, not the people who hang them.
「吊りさがる」 is an intransitive verb. Things are the subject.
「吊りさげる」 is a transitive verb. People are the subject.
But your first TL certainly captures nicely the essence of the original.
Your second TL is way off, I am afraid. The original does not say "on Tanabata".
My own TL:
"For quite a while, (I've been noticing that) mostly things that are unrelated to Tanabata are hanging (here)."