This is somewhat redundant with Igor Pak's P.S., but I found an index for the 1918-1950 range in The Otto Dunkel memorial problem book, starting on page 80 of this pdf.
I found more: Stanley Rabinowitz and others had the same idea a couple of decades ago, and they started compiling indices, not just for the Monthly. Here are Google Books and official links for the two volumes I found:
1975-1979 (published 1999) Website
1980-1984 (published 1992) Website, Errata
Googling "Stanley Rabinowitz 1985-1989" led me to this:
The labor involved in classifying and indexing the 5-year indexes has proved to be far greater than anticipated. We hope to have the 1975-1979 Index out later this year and the 1985-1989 Index is well under way. [Stanley Rabinowitz, June 1996]
I don't know if the 1985-1989 index was ever finished, but presumably there's at least a partial draft out there somewhere. The website http://www.mathpropress.com/ of Rabinowitz's company includes contact information. They say they're looking for authors to publish, among other things, "indexes to mathematical problems or results", but the page apparently hasn't been updated since 2006.