I can't see enough for me to remark on some points, but I think the lower tanks have an issue. It looks like this stand has six 2x4s as columns. (Maybe you have some doubled up and I can't see them, and if so, some of this below is not relevant.)
2x4s (not pressure treated) are rated for ~636 lbs each (compression strength). If all 4 tanks (4x550=2,200 lbs) were resting equally on all 6 2x4s (6x636=3,936 lbs) there would be no problem from a compression standpoint. Provided the tank was properly stable, all of the weight could be safely carried on six 2x4s.
From the picture it looks like the upper 2 tanks are resting on all 6 2x4's, while the lower 2 tanks are resting on 2 2x4s. However, those 2x4s are also carrying some of the load from the upper tanks above as well. If the 2 lower 2x4s (out of the 6 2x4s) were carrying 1/3 of the load of the upper tanks, they would have a load of ~((2x550) + (2x1/3x550))=1,467 lbs, while rated for 2x636=1,272 lbs.
The other issue is that the two 2x4s are set midway at the end of the two lower tanks. The whole of the two tanks are thus "balances" along a wall where the shelf is meant to keep the two tanks perfectly stable and to distribute the weight equally. This will work perfectly when both tanks are completely empty or completely full. But when one is full and the other nearly empty, this will basically become a cantilever and all the stress to keep the one full tank from tipping over will be the fasteners (screws/nails/bolts.)
Now, the side fasteners (screws/nails/bolts) look like they are supporting the platform (shelf) for the lower 2x4s, so to that extent, they are distributing some of the load from the 2 lower tanks back onto the other four 2x4s. I don't know the size or strength of those fasteners, but they would tend to shear if the compression strength of the wood failed. So, that causes me some concern that this could fail. If I had confidence that they were transferring, let's say, 400 lbs off the two 2x4s back onto the other four 2x4s, I wouldn't have a concern. I don't have that confidence especially since many fasteners are not designed or used for that duty: some are too thin, some are not embedded deeply enough, some are not designed from material for that purpose, there may be an inadequate number used, etc.
One obvious "fix" would be to add 2x4s at the midway points under each side of the lower shelf where the lower tanks sit. (Basically, under the midway point of the lower 2 tanks, looking at them, left to right.) This would reduce the compression load on the two 2x4s (by providing two more) and thus reduce shearing stress on the fasteners.
I concur with fwprawn that braces are a good idea much as he noted. Wood is extremely strong under compression, but over time, if the structure shifts even a small amount, under heavy loading, the fasteners will tend to weaken (detach from the wood) or fail (shear.) Braces reinforces the stability and maintains the use of the wood in its most advantageous form and does not try to give too much work to fasteners.
