If the other teams' plan was ever to plunk Craig Biggio so much that he'd make more mistakes in the field, it hasn't worked. In the 259 games in which Biggio has been hit by a pitch and played in the field, he's only committed 12 errors (he's been hit 5 times in 4 games as either a DH, or pinch hitting and not playing the field). Through the end of last season, his fielding percentage was slightly higher in games in which he was plunked - .990 in plunk games and .984 in games in which he didn't get hit by a pitch.
All 12 of his errors in games in which he was hit by a pitch came while playing 2nd base. Back when he played catcher, he threw out 8 of 21 attempted base stealers in plunk games, or 38.1% of them compared to 22.5% of attempts in games in which he wasn't hit by a pitch. Also, none of his 32 career passed balls as a catcher were recorded during games in which he was plunked, so on those day when he did, he wasn't letting any pitch get by him on offense or defense.
Craig Biggio has helped turn 126 double plays in games in which he was hit by a pitch - that's only 21 short of the number of double plays he's ever grounded into - 147. He's only grounded into a double play 12 times in games in which he was hit by a pitch.