One of the local sports columnists (TJ Simers of the Times) was on one of his rants about the Dodgers owner a week ago, and mentioned the Union Station shuttle had been terminated. There are cabs available at Union Station to take you to the game however (I'd guess about $13). Getting back to Union Station might be a trick. I've never noticed a cab stand at the Stadium.
Cabbing from Anaheim Stadium to Dodger Stadium would be the national debt. Assuming no real traffic it would be north of $80. A car service might be around the same price.
If you can plan your trip for off-peak hours, driving is not a bad choice. From Anaheim to Downtown you are essentially taking the same Freeway the whole way (I-5/US101). After the Dodger game, if you can find your way to I-5 (there are signs from the Stadium, but its not intuitive), BUR is an quick drive.