My old Igloo 36qt cooler fits nicely under there, although you could go longer. I just measured it and it's 24"L x 13.5"W x 13"H. I think the crossbars before they start bending are like 28" if I remember correctly. The height is the major thing.
I usually have a heavy rubber mat there as well so the cooler is wedged in there pretty tight. The front passenger definitely has to stand up to get it out. Probably would still need to even without the mat. Depends on how heavy the passenger and how pumped the raft is.
I like having the small cooler on board. It's the beer, lunch & dinner and just rando stuff cooler.
For dry storage on a small raft like that I prefer bags. I really like the big NRS 70L High Roll Duffel Dry Bag. Looks like REI has a killer deal on it right now. It usually goes behind the driver. I can put my air mattress, pillow and sleeping bag all in there for rare overnighter. It's big.
I also have a 20L rolltop dry bag that I like to use for spare clothes or whatever. It's pretty easy to get into. I often keep it under my (rower) seat. And then I really like the OPST rolltop waist bag. It's great for swinging and has the added benefit of being rolltop water proof. You probably see the trend here.
If it was a big raft then sure, I'd have a nice dry box, but on a small raft like the Striker I think flexibility is the key. And soft bags are, by their nature, more flexible.
I do use Rubber Maid Roughneck tubs sometimes. Those are 24"L x 15 5/8"W x 13"H so could also fit under there, but the 15 5/8" width means they'll be hitting the front guy in the calves or in the rower's feet. If the rower is short, that might actually work well. I'm not short so not for me.
I usually only use the tubs when I have the rack on, which is only when I have the frame on. Or if I'm solo mode (check your manual), I'll have a tub or 2 up front.
Good luck!