Water – Why is there moisture under some new wall plates in the basement

Multi-part answer here:

1. It could be the water table, in which case, there's not a lot you can do. If it's the water table, then you'll have water seeping up through the floor. BUT consider that when they built the house, they dug a bowl in the ground, and then backfilled with loose soil, rocks and whatever other garbage they had lying around. You could have a permeable (gravel and sand) fill inside a relatively impermeable bowl (clay). In this case, sloping the ground away can help. It will at least get rid of standing water up against the foundation and help mitigate anything flowing through the walls.

2. The major worry is mold. Make sure the area is clean, so that there's nothing for mold to feed on. You can spray with a biocide if you see any growing, and certainly don't enclose it by adding drywall over the area.

3. If you have rising damp, then an internal sealant won't do anything. the hydrostatic pressure will force water through. A dehumidifier will remove the symptoms.

I would definitely be concerned, mostly about mold but also about property damage (you said it's used for storage) and long-term foundation damage.

If your downspouts are not properly discharging water away from your house then you definitely want to address this. In the grand scheme of wet basement solutions, this is probably the cheapest fix you will find. Tight property lines or not, make sure that your downspouts discharge water far away from your foundation. It's hard to say whether you would benefit from a dry well.

If this fails then you should investigate whether you have a French drain and/or weeping tiles and whether that system has failed. If it is just clogged it could be a somewhat easy repair. If you don't have one or it has failed beyond repair then you will need to install or replace the drainage system. You said your house has tight property lines so it might not be realistic to dig up the outside, in which case you'd have to look into an interior French drain and sump pit solution. Either way this is probably going to be expensive.

I would definitely advise you to bring an expert in to review your situation and suggest some solutions. Make sure to get a few opinions/quotes. Fixing this this is probably outside of the DIY-scope for most people as messing with your foundation is risky business.