As a spinner, there are ways to stop batsmen going after you (and likely paying for trying). First, obviously, you need accuracy, and loads of fingerspinners have that. Accuracy alone is not enough, however, and there are a few other things you can use to compliment it and use to stop batsmen attacking you (obviously you won't be able to get it right every time, and obviously every now and then batsmen will fluke it when having a swing, but a good bowler will get good figures more often than not).
The best method is to turn it big; however, only wristspinners can do this on non-turning pitches, as fingerspinners don't have the tools to do so. Hence, if a "normal" fingerspinner plays on a non-turning pitch, it's usually very easy to score off him. And of course, the problem is that wristspin is usually very hard to control, so only very special wristspinners (namely Warne and Murali) have harnessed the control and the turn neccessary for ODI success on all surfaces. Many fingerspinners and non-big-turning wristies have had success on slower or turning pitches but not on quicker, less dry ones.
Another method is to bowl consistently very fast and flat: Dharmasena and, until the 1999 WC, Kumble, excelled at this. They usually bowled at at least 60mph, sometimes even quicker, making it hard to come down the pitch to them, but they bowled so flat they weren't often easy to cut either. However, few fingerspinners have ever managed to do this.
Another method, more difficult still, is to bowl a Doosra: Saqlain and Harbhajan did this. Now, though neither Off-Break nor Doosra will turn much on a non-turning pitch, in a game where batsmen are often looking to attack this is often still enough. Saqlain and Harbhajan actually ended-up with very different careers, Saqlain being a big wicket-taker and only reasonably economical; Harbhajan being generally very economical but not a huge wicket-taker (despite occasional sensational hauls). Of course, Saqlain and Harbhajan to boot with their Doosras were skilled flight bowlers, varying flight and length well.
Something Daniel Vettori perhaps does even better still - and he also possesses a borderline sixth-sense and seems sometimes to know better than even the batsmen what shot he's going to attempt. A good thing, too, as Vettori doesn't have a Doosra (though he does of course have an excellent Arm-Ball).
Aside from these, there have been no successful fingerspinners in modern ODI cricket, except possibly Carl Hooper, who I never saw bowl much and don't know the secret of his relative success.