I've folded up more than a few steel frames/forks. There was usually a car or a loose mutt involved, but going ass over tin cups in a crit did one in and another died in a road race stack up.
Most of the stuff built today has a CF fork anyway. And let's not get into the fun and money of replacing shifters, ground up rear derailleurs and those snapped off hangers.
In the serious ****, my medical expenses, sheets ruined with blood stains, re-stocking the medicine cabinet and race day bandage kit, missed work and shredded clothing usually equaled or exceeded the bike repair or replacement cost anyway.
If money is an issue it might be best to re-think racing a bicycle. Especially in the demo derby most amateur and beginner level crits are.
If the OP is just asking if Ti is good for crits...sure it is. Depending on the specific model, Ti can be a bit flexible, but there are some Ti frames that are plenty stiff enough for a big, powerful sprinter types. The same goes for aluminum, steel and carbon. The individual choice of frame will matter more than the material.
The same goes for money. A cheap carbon frame that would be perfectly at home on the crit circuit can be had for a grand or slightly less. Hell, I would willingly put a leg over one of those Nashbar 105 carbon bikes that sell complete for around $1,200-$1,500. Slap on the race wheels and I would be at no disadvantage on the starting line. And if it gets toasted and an entire new machine sans race wheels won't kill the bank account.
By way of comparison, try finding a $1,500 Ti, frame with a CF fork for around that same money. The choices might be a bit limited.
Aluminum, especially a good used aluminum frameset, can be had very reasonably these days. They are mostly more than up for the task of a summer or two of killing it and drilling it around a 2-city block square course with guys that would slit their own mother's throat for a $5 trophy. They may offer a more jarring ride and the, again, they may not...but my backside can put up with almost anything over a 45-60 minute sufferfest. Adding a CF seatpost, the right set of wheels and some good bar tape can overcome a lot of that too.
As long as the geometry and fit are solid, the frame material is secondary to me.