Correcting for multiple comparisons between two sets of paired groups I am looking for the most appropriate correction for multiple comparisons in this situation. For two classes A and B we have N groups, for example:
A = {1: [1,2,3,4,5 ...],
     2: [5,2,4,4,9 ...],
     ...}
B = {1: [10,20,30,40,50 ...],
     2: [22,32,45,56,11 ...],
     ...}

The comparisons will then be between the means of A1 vs B1, A2 vs B2, and so on. The elements in the respective Ai and Bi series are not paired.
Right now I am simply using t-tests for each comparison and then applying Bonferroni's correction, but I am wondering whether there would be a more appropriate strategy here.

The problem with Bonferroni is that you won't detect any violations of the null hypothesis if $N$ is becoming large.
Thus, instead of asking for one test that covers everything, you should change your goal a bit: e.g.  the Benjamini–Hochberg procedure answers the following question, which might be useful for you:
Which tests, out of my complete set of $N$ tests, should I consider to significantly reject the null hypothesis, if I want to have only a ratio of $q$ of those to be wrong, for some $q$ that the user can provide?
E.g., if you choose $q=0.1$, then the Benjamini–Hochberg procedure (BH) gives you those tests out of your total $N$ tests that are the most significant ones. Let's say (BH) returns e.g. $k=50$ such tests to you. Then BH makes sure that on average only $q\cdot k=0.1\cdot 50=5$ of those tests will be wrong.
