1) It's not Biden's job to prosecute Trump, that's up to the legal system on the state and federal level to take that case, which are independent of the President.
2) The only way Biden can help Trump is if he pardons him, which he is extremely unlikely to do (remember Nixon got pardoned by the VP of his own party and that was so controversial that Ford had no chance of being elected after completing Nixon's term).
3) Personally leading and calling for prosecutions agains Trump would be construed as "lawfare" a la "Special Prosecutor to appoint Hillary" it's a very dicey thing even if Biden has the authority to do it. It cannot have the appearance of a succeeding candidate attacking his predecessor.
4) Progressive Democrats are no more likelier to bring actions against Trump if they became POTUS because of 1) and 3).