Grogan Stump, popular Minister for Magic and Tutshill Tornados fan, founded this Department during his tenure.
The exact year is unknown, but Stump was Minister from 1811 until 1819, so the date would be somewhere within that time.
Two dates are given in canon for the formation of the Department of Magical Games and Sports. Quidditch Through the Ages gives the year as 1750 when listing the rules the Department laid down (QA6). Rowling's detailed list of Ministers for Magic and their accomplishments on Pottermore, however, says that the Department was established by Grogan Stump who was Minister for Magic from 1811 to 1819 (Pm).
I wonder if the Banchory Bombers' Statute of Secrecy-breaking antics which resulted in their permanent ban in 1814 might have had something to do with this. Perhaps that incident revealed the need for a stronger, more proactive Department of Magical Games and Sports than the one which had been created 60 years before, and Stump spearheaded a complete revamp of the Department as a result.