From an architectural specification perspective, we usually bolt them to concrete/pavement, cast them in, or just set them there. Generally speaking, when you have an ~8' long thing built out of metal it's not going to be moved unless someone really wants to, and if they're going to bring means to abscond with it, they're going to bring means to cut it free of any moorings. Particularly if it's got some wobbly things awkwardly affixed to it.
I believe the current bike rack was built in house 3-5 years ago, and hasn't been stolen or tampered with yet. If we needed to affix it to the grass I'd look at driving in several hairpins of #6 epoxy bar to ~24" (if it's not already), but that seems silly.
To make it a bit nicer, I'd look at a sand bedded packed gravel pad for less mud and ease of shoveling.
-metis