I was wondering if after a year of growing cycle my basil will perish while my mint plant will go on indefinitely?

You might be on the south side of frosts. I would expect some years your basil might survive. Other winters not so much.

Your mint will most likely go on. Basil is generally an annual.

Not sure what your growing zone is like, but basil tends to be an annual--it wants to set flowers and then stops growing.

Mint is very hardy. It does does flower, but persists because it grows vigorously by a very tenacious root system. Even if a branch touches the ground, it will try to send down roots. Doesn't require much watering or care.

The only basil that does not seed, but requires 80's to survive if you keep the flowers pinched back, is African Blue Basil. 2) thai basil, 3) lime basil