Bruce has them in locked filing cabinets, tucked away in a corner of the trophy room, each with vaccuum-sealed, better-than-archival-quality sleeve and board. He keeps a key to these cabinets on him at all times, whether as Batman or Bruce Wayne. He also scans a copy of each picture into the Batcomputer, under heavy encryption.
The first such piece of art was donated by a 7-year-old girl, who, on a rather dreary August afternoon, entered the Gotham City Police Department. She insisted on seeing Commissioner Gordon, refusing to speak to any other police officer, and when she eventually did, extracted a promise that Gordon give the drawing to Batman the very next time he saw him.
When Bruce got the drawing, he tracked down the girl and gave her a spare utility belt (empty and no special functions). 10 years later, the girl also found herself the unlikely recipient of a full college scholarship from the Wayne Foundation after being nominated by an anonymous party.
Since then, the GCPD now has a box in the mailroom for children's art, which Batman regularly checks. While they don't all get utility belts and scholarships, each does get a thank you note signed with a hand-drawn batsymbol.
To date, 7 such boys and girls have grown up to become criminals. It breaks Bruce's heart every time.