East Side Access will certainly increase Manhattan-bound LIRR capacity, but I don't think it will do much to improve matters on Long Island itself. Both the Main Line and the Babylon Branch are pretty much at capacity during rush hour, with the Main Line having to run trains in peak direction on both tracks. The key improvement that is needed there is a third track from Queens Village to Hicksville, which would allow for more trains to run to fill that new East Side tunnel, and also for more reverse commute service and so on.
The other thing that the LIRR needs to do is expand electrification. It has done wonders for the Ronkonkoma Branch, so why not extend the benefits elsewhere? I think the top priority should probably be Port Jefferson, to be completed when the third track is, as that line has the highest level of diesel train service. Oyster Bay is another good candidate, as that branch is not doing very well, partly because diesel trains are slow and its route is not very direct.
Also, the LIRR could reclaim some of its abandoned lines, such as the West Hempstead-Country Life Press-Mineola connection for cross-island service, or use the right of way eastward from Garden City on the old main line.
But the real problem, I think, is not the LIRR service, which is pretty darn good for commuter rail. It is in getting people to the stations, which currently involves massive parking lots and lots of cars. And while there is a Long Island Bus, I think the MTA should seriously look into building streetcar lines, specifically to connect to LIRR stations, and reduce the demand for parking.