This is not a complete answer to the question, and I am not sure that a complete answer is written down anywhere in the literature (I tried to adress all the 10 comments that my previous answer recieved).

Let us notice first that already the topological classification of non-compact surfaces is quite a delicate subject. It is desribed for example in the paper http://www.jstor.org/pss/1993768, 1963, ON THE CLASSIFICATION OF NONCOMPACT SURFACES, IAN RICHARDS. In particular a new phenomena apears here -- a certain ideal boundry of the surface, that is a totally disconnected, compact separable space. I guess a good illustration will be a disk from which we throw away a Cantor set on the x axis.

Let us give now some examples that illustrate the phenomena that happen for non-compact surfaces, if we take in account the symplectic form. First of all there is the simplest case when the surface has finite topological type. In this case we have two  topological invariants, the fundamental group, a free group on $n$ generators plus the number of punctures (or boundary components) $m$. In this case a complete classification of symplectic forms can be given.  Either the surface has a bouned are $A$, in this case this area is the only invariant. Or it has an infinite area. In this case there are $m$ types of surfaces. Namely for every boundary (or puncture) we can check if it has on open neighborhood that his finite area, or not. The number of components near which the area is unbounded can be any between $1$ and $m$.  

Example. Consider $S^2$ with two delited points. Then either it has finite area, or it is symplectomorphic to an infinite cylinder $S^1\times R^1$ with the form $ds \wedge dt$, or it is symplectomorphic to $R^2\setminus 0$, with the form $dx\wedge dy$.

If the number of punctures is countable, and every puncture has a neighborhood that is diffeomorphic to a punctured disk, then the situation should be very similar to what I have described above. Namely 1) the area can be bounded. 2) The number of UNBOUNDED punctures for wich every neighborhood has infinite area is bounded, in such surfaces are enumerated by natural numbers. 3) The number of unbounded punctures is infinite, in this case we just need to cound the number of bounded punctures, that can be finite of inifinite. 

Now, my impression, that if we will take more complicated examples then what will happen -- the symplectic form will induce a certain structure on the ideal boundary. It is not quite clear what structure this will be. Maybe this will be just a closed subset of the ideal boundary, corresponding to points for wich every neighborhood has infinite volume. Maybe something even more complicated. This is just a speculation.

ADDED

For the simple examples that I have descirbed the proof should be ALMOST identical to Moser's argument. Indeed the simplest examples have the following property:
these surfaces can be decomposed in a countable union of compact pieces, all of which apart from one piece are annuli, and one piece is a compact surface with a boundary. Now, on a compact surface with a boundary (as well as on the cylinder) the symplectic form is uniquely defined by its area -- this can be done by Moser argument (we need that in a neighbrohood of each boundary the symplectic form is strandard, which is automatic in our case). Now the symplectomorphism can be constructed inductively.

Consider for example the case of $R^2$ of infinite area.  We can take any exostion of $R^2$ by cylinders. For example we indroduce some coordinates on $R^2$, and conisder cylinders of radiuce $n,n+1$. We don't care what is the exact expression of w. The crucial thing for us is that the sum of areas tends to infinity. Now we take the standard $R^2$ and take a decompositon in concetnric cylinders of needed area. Then define the symplectomorphism from the standrd $R^2$ cylinder by cylinder. For the class of surfaces, that I dealed with same thing should work. Though I am not sure that this is the best "proof" of the statement.