A mitzvah for an aveira? or at someone else’s expense?

I heard this story from a cousin of mine (a bochur) who lives in Eretz Yisroel. He said that he was sitting on a bus when an elderly woman came on. The bus was crowded and there were no seat lefts. There were a lot of women and girls on the bus. He said that he wanted to get up to give his seat to the old lady, however he didn’t want to ‘touch a girl’ (he should do a mitzvah for an aveirah?), so he stayed in his seat.

I’m very curious to hear comments from others about this situation.