Here are a few more options for you. “Homines” is just a bunch of people, not really the concept of “mankind,” so my changes to the good suggestions so far mostly revolve around that. Probably the most literal translation would be something like this:

Humanum genus vitia humanum generis vincet

literally ‘the race of men will conquer the ills/vices of the race of men’

Sometimes you see the word “mundus” (lit. ‘the world’) used in poetry to mean ‘the inhabitants of the world’. So if you want something more poetic and concise, plus some nice alliteration:

Mundus vitia mundi vincet

lit. ‘the [inhabitants of the] world will conquer the ills/vices of the [inhabitants of the] world’

Another option would be to put the verb in the subjunctive, as many mottos do, so that the statement is more of a wish than a statement of fact about the future. This simply involves changing “vincet” to “vincat.” So “Mundus vitia mundi vincat” would mean “Let/May mankind conquer the ills of mankind.”