It is generally accepted that in the application you are describing a form of closed cell Polyethylene should be used. The common name is Ethafoam originally made by DOW. That name and line of products was purchased by, and is now manufactured and distributed by Sealed Air (the bubble wrap people). Versions of the same basic product with just slightly differing characteristics are made by other companies and go by names like Polyplank (Celluplank for example was Sealed Airs version before they purchased the Ethafoam name from DOW).
Occasionally you will find other types of polyethylene foams like crosslinked versions - minicell and plastizote are examples - but it is expensive and pretty uncommon.

The other common category of foam used for art crating is urethane foams which are supposed to be less chemically stable (prone to hazardous off-gassing) than polyethylene.

It is worth noting that since 1992 when the Art In Transit conference was held (along with the resulting publications) there has been an emphasis on limiting or slowing the rate of temperature change that paintings are exposed to. The Urethane foams are supposed to provide a higher degree of thermal insulation than Ethafoam (especially in the amounts normally used - Ethafoam is typically used in strips while Urethane is often used to line the whole crate. If there are concerns about thermal insulation when the crate is being traveled some kind of insulation may be required. While considered exceptable in that application. Urethane foams are generally not used for storage. In a single crate situation Ethafoam is often used in combination with Polystyrene (styrofoam - the type used construction not coolers) which is chemically more stable and will provide insulation properties during transit.

Another not-uncommon scenario can include the use of both basic categories of foam. When double-crating a painting the inner crate may have Ethafoam - often one inch thick (enough to allow for handling the frame) and then a Urethane foam for its dual cushioning and insulating properties between the inner and outer crates.

Two other significant issues related to paintings being left in crates are vapor barriers and the practice of sealing the crates.
The first part involves providing an effective barrier between any potentially hazardous materials in terms of offgassing (plywood) and the probably represents a greater concern than even than even that of the foam. Something like marvelseal is recommended under these conditions. That is especially true if paintiings will be sealed in the crates.
By creating a micro environment any pollutants being formed will end up being concentrated over time. The natural results of the ageing of the wood stretcher/strainer and frame - or with paintings even solvents remaining in the paint itself can potentially pose a hazard to the paper for example.
With paper obviously if the glazing package, or "passe-partout"is top notch these concerns are minimized.

In situations where artwork (often due to size restrictions) is kept in crates in a stable storage environment I have seen the lids of crates tightend down on a series of evenly thick shims creating a narrow gap to allow for ventilation. For someone who is really concerned, you can even remove the lid and attach it to the back of the crate (keeps it associated with the crate as well as keeps it from warping over time). If you do either of these you would also normally include a "breathable" dust cover.
Hope that helps or at least is semi-coherent!