After filtering down my query, I think I have the final list. There are 205 players who went from sub-par at eliciting a base on balls to better than average after a substantial trial at the major-league level (min. 1000 plate appearances) and who continued to draw walks regularly. They "learned" how to induce a walk.
The first player that I looked for on the list was Sammy Sosa, who went from an average right fielder to a Hall-of-Famer in part because he became a bit more picky at the plate. At age 29 Sosa drew 73 walks, a then-career high, 28 more than the previous season (and only 5 more intentional walks) as he saw his home run total go from 36 to 66 in just one more at-bat. Sosa's first season produced a line of 33 walks to 150 strikeouts. His career high in walks came in 2001, arguably his best season, with 116 walks (37 intentional) against 153 K's.
Sosa's 26th on the list. Ahead of him are a number of Hall-of-Fame- or near-Hall-of-Fame-caliber careers. Most of the better players changed their walking ways early in their careers. Most enjoyed more success after doing so.
John Jaha, for example, went from an offensive sink hole at 1B/DH to, briefly, one of the better power hitters in the game. Ken Caminiti went from a good defensive third baseman to a great offensive one at the age of 32 with a little help from his friends, the base on balls and anabolic steroids. Don Mattingly briefly resurrected his career at age 32 when he went from 39 walks in 686 plate appearances in 1992 to 61 in 596 plate appearances in 1993 while upping his home runs and all of his ratios (and his park-adjusted OPS went from 108 to 118).
Others turned to the base on ball in as refuge in a poor season. Some were repaid with greatness in future seasons (witness Richie Ashburn's sub-par, walk-laden 1952 followed by much success while continuing to draw walks). Some (e.g., Jose Hernandez) turned to the base on balls but still witnessed other aspects of their game shutting down.
Here are the top fifty turnarounds in walk-ability. For each the age at which the went from a sub-par to a superior walker is listed along with the year, their previous walk shortfall below average expectations ("Pre"), previous plate appearances, their post walk surplus ("Post"), the post plate appearances (including the season listed), and the walk above expectation difference before and after. They are ranked by the walk differential: