I have a basement that is about 2000 square feet. It will be a 3 bed 2 bath apartment and has a separate furnace already. I want to put it on a separate meter from the rest of the house and wondered what the approximate cost would be?

In Manhattan, Malibu, or some little backwater rural town? Labor costs play a large factor in this as do trouble and cost for permits, materials, etc.

Best answer (as in closest) you would get is by calling at least 3 local electrical contractors and get written estimates from them.

Was this a permitted remodel? Some utilities will not do a separate meter install for a non permitted apartment. The utility company may not charge for another meter but after that you need to pay. Without knowing the amperage, length of run etc. it is hard to give a price. You will need a breaker box, outlets and lighting to code. As Smoothy noted, get 3 estimates from licensed electrical contractors but I would guess something in the $5,000 range depending on needs..

You have a lot going on here. Typically, for a single family residence you are allowed one meter. But you are no longer a single family residence because of the apartment.

I suggest you contact your utility company and see if they will install a second service to supply the apartment or if they need you to apply for a multifamily residence permit before they will run a second service to the residence.

It's probably not worth it. If you live there, average out the cost and add it to the rent under a "utilities included" type of agreement.
If there is a separate panel for the basement it will be fairly easy to figure out the cost using one of the many energy monitors available.
If both levels use the same panel, just watch the bill. Others have mentioned the legality issues... have you looked into that? It's not hard to add a panel and a meter socket, re-wire the existing circuits and separate the levels... but it 'ain't' cheap.