To be honest, a pick-up service will probably charge just as much as one of the rip-off cabs. I've just looked online and it'll probably cost you 15-20 Euro, but I've sure there are cheaper ones. There are plenty of them around if you look (try googling 'taxi bucuresti airport') but I've never used one so I wouldn't like to make a recommendation.
Taxis from the airport aren't as bad these days. It used to be that one company had a monopoly and they charged three times the rate of regular cabs. Nowadays any cab company can go there to pick up clients (as long as they have paid the special tax for entering airports). The last time I got a cab home from the airport (in Jan this year) it cost me about 35 RON (8 Euro). I do speak Romanian though, not fluently, but well enough that the driver knows that I know where I'm going and won't put up with any 'scenic' routes.
Some tips on airport taxis (and Bucharest taxis in general):
1. Check the rate displayed on the side of the cab. It should be around 1.4 ron/km (cost/distance rate) and about 14.0 ron/ora (per hour waiting rate). Expensive, but legal cabs charge 3.5ron/km. Criminal cabbies have 14.0ron/km on the door knowing that drunk/tired/foreign clients might misread it.
2. Use one of the more well-known companies like Cristaxi, Taxi2000, Mavi, Cobalcescu, Leone, Meridian.
3. When you get in, check that there isn't already something on the metre (from the client they have just dropped off at the airport).
4. Check the meter. You should see the total for that ride (suma:) and the rate that is being applied (1.40 if you're moving, 14.0 when you're not - or whatever the rates on the door were). The crooked cabs will set their meter to use a higher rate than the one advertised on the door.
5. Have a local friend send you something you can write clearly on a piece of paper giving the address, sector, and some landmarks (closest main road, for example). Give this to the cabbie when you get it.
6. Make sure you have plentry of change so he can't pull the old 'sorry mate, I can't give you change for that 100 note' bullshit, hoping you'll just let him keep it all even though it's double the fare.
7. Don't accept an offer for a 'cheap cab' from the dodgy dudes hanging around at the arrivals lounge or near the exits. They won't be cheap and they might rip you off. Only use cabs from the official taxi rank in front of the arrivals building.