Rex Caldwell, a playoff loser last week, birdied the final hole to complete a four-under-par 67 and take a one-stroke lead yesterday in the rain-plagued third round of the $350,000 Phoenix Open golf tournament.

Caldwell's 20-foot putt through a steadily drizzling rain finished off his eighth consecutive round in the 60s. It gave him a 54-hole total of 201, 12 strokes under par on the 6,726-yard Phoenix Country Club course that was spotted by pools and puddles of water.

"You just try to shoot as good as you can. I did the things I had to do when I had to do them," said Caldwell, who was deprived of his first tour victory last week in a playoff.

His go-ahead birdie putt gave him a one-shot margin over Bob Gilder (66--202), a former Phoenix Open champ and a three-time winner last season . . .

Pat Bradley shot a second straight 69 to lead by two strokes over Beth Daniel, who had a tournament record 64 in the third round of the $150,000 Mazda Classic of Deer Creek, Fla., the opening LPGA event of 1983.

Bradley's bogey-free round gave her a 10-under-par 206 going into today's final day . . .

The 1983 U.S. Open golf championship will offer $500,000 in prize money, an increase of $125,000 over last year, the United States Golf Association (USGA) announced in Colorado Springs, Colo.

First prize in this year's 83rd Open championship, to be played June 16-19 at the Oakmont (Pa.) Country Club, will be $72,000, $12,000 more than Tom Watson collected as champion last year.

Prize money for the U.S. Women's Open was increased from $175,000 to $200,000.

The USGA also announced Jay Sigel has been named playing captain of the U.S. team for the 1983 Walker Cup match against amateur golfers from Britain and Ireland.

Other members of the 10-man team that will travel to the Royal Liverpool Golf Club in Hoylake, England, May 25-26, will be named in the near future, the USGA said.