I lose the flavor of basil when I make pesto sauce
I don't have a definite answer, but here are a few thoughts on things that might help.

First, are you drying the basil after washing it?  Shake out the extra water, and pat it dry, maybe let it sit for a bit to evaporate.  The extra water will dilute the pesto a bit, and not be good for the flavor - and if you think the problem might be too much water in the basil, no need to make it worse with extra water on the surface.

Also, are you adding basil stems?  Depending on your variety of plant and the steps you take in your recipe, the soft stems or tips might have more flavor that you're loosing by leaving out, or more plant matter to dilute your basil paste if you leave them in.  Taste the stems, and check your process, and consider if it would help to take the stems out/leave them in once and see if that helps.

Also, it might help to check your salt content.  Some pesto recipes use salt, others use salty cheese, I don't know what your recipe calls for - but salt brings flavors out, and one symptom of under-salting is "watery" flavor, so you might just need a bit more.  Adding salt might help your basil paste taste like "basil" instead of "grass".

I was wondering, given your comment on seeing a lot of water on the cutting board from the basil, if you might be losing flavor from the basil juice when cutting.  Preparing the basil on the cutting board seems the method most likely to have this problem, but even if using the mortar, if you crush ingredients separately then mix in a different container, you might be losing some of the flavor in juices.  On the other hand, most blender recipes (and some mortar and pestle ones) make the pesto in one container, which means the flavor should still be in the mix and not lost- so that might not be the problem, or at least not the only problem.

If the basil is just too mild for your tastes, or has a higher water/flavor oil concentration than you'd like, you might be able to address this by letting the basil dry a little, maybe till wilting, before making pesto to concentrate the flavor a bit.  You might also consider adding some dried basil to strengthen the flavor - I would think completely dried basil would not give the right consistency, but a mix of fresh paste and dried flakes might give the flavor profile you're looking for.

If the basil tastes like the basil you expect when you take a leaf and taste, and grass after it's made into paste, then the flavor must lost somewhere along the way - then you can look at your process and guess where the flavor is going (like sap left on the cutting board, or whatever).  If you taste the leaf and find it doesn't taste basil-y enough to begin with, then it might just be you prefer a stronger basil or a different variety, and doctoring your basil paste with flakes (or even other herbs/spices) might be the only way to get the pesto you want if no other basil varieties are available to you locally.