If they turned Manhattan into a prison, why didn't they just blow up the bridges instead of mine them?
It would give prisoners hope (albeit false hope) with the bridges there.
They might want to use the bridges again in the future. Maybe after the crime rate declines, the administration might want to go in and "clean house" (the US has become a totalitarian state and they might not think twice about just killing all the prisoners to get the city back), and the city would be easier to get to with bridges.
In the novelization, the city was originally abandoned because a massive nerve-gas attack made it too toxic to expect anyone to live in willingly. The gas residue presumably won't last forever; give it a few decades for weather and time to get rid of it, and they'll want to re-claim the place.
Also, if there was an emergency (such as a particularly massive uprising) that was threatening to spill over to the mainland and the United States Police Force needed to get into the island en mass to deal with it quickly, then it'd be easier and quicker to ferry a large force across the bridges and through the tunnels than just relying on an air or naval assault. They'd know where the mines were so that wouldn't be as much of a problem and could push the ruined cars aside with tanks and heavy vehicles, but if they'd blown the bridges / tunnels they wouldn't be able to get in.
Minor nitpick, but how did the President's escape-pod drop thousands of feet and crash through a brick building, yet wind up perfectly upright?
It's probably designed to do just that.
Or, since the hatch was on top, the prisoners turned it upright when they found it to get it open.