I don't know about on the Isle of Wight side, though I was on the island over the weekend the other day. I highly doubt trains before 5pm will be too packed. Even the Soton to Pompey services I didn't travel on much, but I'll tell you what I know:
The GWR trains will certainly be busy during the morning and evening peaks, and may also be busy if there are events along their routes. They're usually just about bearable for getting a seat at other times, but do generally get very busy if anything is slightly out of the ordinary.
The SWT services obviously start from Southampton and they're slow stoppers, so you should certainly be able to get a seat. They do take forever though, but bear in mind it's only a double track line so they don't get overtaken — if your next train is an all stations stopper to Portsmouth, it'll still be faster than waiting for the one after! One thing that might catch you out though is that for most of the day they terminate at Portsmouth and Southsea due to (I believe) not enough platforms at Portsmouth Harbour (they would have enough but one of the piers is dodgy so they closed it decades ago...). You'll have to change, I'd recommend doing so at Fratton because it's a much easier connection than at Portsmouth and Southsea, unless you want to see the pretty terminus station in stark contrast with the ugly through platforms!
I think it can at certain times be quicker to get a Southern service and change onto a train coming from Eastleigh, but if you've been reading the news, you'll know not to count on this!