Re: Foam Core VS Balsa Core - no stringers
Well, since no one except ondarvr has mentioned composite design and what to use for cores, I'll cover it a bit more.
Wood deck
There are two ways to build part of a boat (like a deck) with a core. The first way you use a strong material like plywood and waterproof it. This means the plywood is the strength of the deck. The glass over it protects it from the one thing that will destroy it quickly... water. If the wood is protected from water coming in, it will last longer than you own the boat.
Unfortunately most people with wood decks either let them get cracked or screw or bolt things onto them without sealing the holes (seats for example) and this lets water get to the wood.
The problem with your balsa deck wasn't the balsa, it was the water that got to it.
Composite panels
The second way to build something with a core is as a composite panel. The difference between the wood deck and composite deck is that the strength of the composite panel comes from the skins. Google "composite panel" if you want the full explanation, but basically the core material holds the skins apart and parallel. It's like the web in an I-beam.. the top and bottom hold the weight, the web keeps them in place.
Don't mix up the two concepts. You can't take a plywood deck, replace the plywood with foam, and get something equally strong. Foam isn't stiff/strong enough to work like the plywood deck, so to replace ply with foam you have to alter the deck construction to make it a composite panel (you have to thicken the skins and change resin).
How composite panels work
A composite panel is essentially two high strength skins separated by foam. The thickness of the foam affects the stiffness and strength of the panel. Holding two suitably strong skins an inch apart makes the panel about as stiff as inch thick plywood, but lighter. Aerospace engineers call this design form "stressed skin construction".
Fiberglass is plenty strong for the skins, and you can also use fancy fibers like kevlar or carbon if you need more strength. The fibers are only part of the panel, though.
Core requirements
The core has to be strong enough to hold the skins in place. Specifically the glass must adhere well to it, and it must resist crushing and shearing (must not crush or pull apart). If the skins pull off the core (called delamination) then the deck is flexible and weak (and won't hold your weight) until repaired.
The problem with all the foams sold in home depot and the like is that they're low density insulation foams. That means they're very low density because they're meant to block heat transfer. That also means they have very low compression strength and low shear strength.
The core in a composite panel has to A) Keep the skins from moving away from it and B) Keep the skins from crushing it inward. If you think about it, the skins can't move any other way in normal use. To keep the skins from moving away the foam core must not break apart (shear strength) and to keep the skins from moving inward (being crushed) the foam core must have compression strength.
Core foam specifics
To give you an idea, the foam normally used as a composite core material is usually 8 lb/cf or better density.. it's like a soft rock in hardness. You can't easily dent it with a fingernail, and a 4 inch thick piece won't crush under your weight. These properties make it a good composite core. I'm sure you know how easily insulation foam crushes by comparison. Most insulation foam is between 1 and 2 lbs/cf density. They're both called "foam" but that's about the only similarity between them.
Balsa core
Balsa, specifically end grain balsa, is also an excellent core. It's fairly light, and if it's in sheets of squares bound together with a glass mesh it conforms to curves well. It's very, very strong under compression, stronger than foam. Hence it makes for light, stiff decks and structure. Again though, the problem with it is that if water gets to it, it rots.
Poly resin
It should be obvious from the description of how composite panels work that the skins have to be glued to the foam very strongly. If the foam and glass don't break, then the panel will hold until the glue gives out.
A composite panel made from high density foam with glass skins and poly resin will be strong and stiff, but the weak link in this case is the resin. Poly is an ok glue, having about a 200psi adhesion strength but if the force on the panel (deck) exceeds that limit, then the panel delaminates by the core separating from the glass or the individual glass fibers pulling loose in the skins. That wouldn't happen in a solid piece of fiberglass unless it bent a lot... composite panels work differently.
Epoxy resin
That's an area where epoxy shines... it's a great glue with 10x the adhesion of polyester resin, so epoxy composite panels are very strong. There's even a couple marine architects who design boats for epoxy composite construction for this reason... their boats are easier to build and lighter than any molded polyester resin boat. The hulls and structure have plywood or foam cores.
Summary
So, to recap... if you build any load bearing structure out of cheap foam sheets with glass over them, it'll fail pretty quickly. The foam is just too light. You can use the foam for flotation (since it's low density and light) or in composite panels or filler for certain non structural applications (T tops no one walks on or similar) but if you want a non rot deck you either need high density foam sheet (expensive) or an alternative like polycore.
Polycore (the major brand is Nidacore) is a plastic honeycomb material made for use as a composite core. You can also get Nomex and aluminum versions. I'm making a deck now out of it using epoxy, with two layers of half inch core and about 12 layers of glass. It'll be lighter than plywood, stronger, and more impact resistant. It'll also never rot, but I'm mostly doing it for strength and weight reasons.
If you can't stand to have a wood deck, pay the price and use a commercial marine foam product. Expect to pay anywhere from $150 to $400 per 4x8 sheet. I pay about $60 per 4x8 for poly core. Some folks have also tried "casting" their own foam sheets using pourable foam, which comes in densities up to 16 lb/cf.
Or, just relax and use a plywood core (you can use plywood as a composite core, by the way.. use thinner ply than 3/4" and thicker skins) and take the time to waterproof it. Seal all the holes and keep them sealed and you'll be fine.
Erik