Each corner of the stand will be made like this. In effect you are gluing and nailing 2 x 4's together to make 4 x 4's, with notches formed by the sandwitch, to hold the cross pieces. This carries the weight of all of the tanks straight down to the floor, so no joint holds the weight. You can use deck or dry wall screws instead of nails if you wish. All those do is hold the boards together while the glue dries. Every place wood touches wood, you use glue, and that glue can be: (from Home Depot or Ace Hardware)
Then use small nails or screws to attach a panel of 1/4" thick plywood or hardboard across the back to provide resistance to "racking", plus glue again. The shelves for the tanks could be 3/4" plywood, with the corners notched to fit around the vertical corner pieces. Those can be screwed or nailed on - no glue needed.
I think you could use 1 x 4's instead of 2 x 4's and this would work just as well. Someone else might want to comment on that. Incidentally, that sketch was done fast and isn't to scale at all.