I have noticed that basil changes its taste as it ages, but not as to get bitter. On the other side, I like a bitter taste, so I often overlook (overtaste?) bitter notes to which other people show a strong reaction. Try getting some fresh basil and comparing the taste/aroma of the fresh and the old one, maybe it is really just the aging difference you perceive as unpleasant. I regularly use the stems when I have fresh basil.
Generally speaking, you cannot deceive your brain to not perceive bitterness (unlike sourness, which is counteracted by sweetness - the pH value of Coca Cola (2.6) is lower than that of tomato juice (4.1), but the soft drink is perceived as sweet, not sour). But if you mix it with another basic taste, the combination might feel much better than the pure bitter taste. The other strategy would be to just dilute with something bland, in order to reduce the bitterness intensity. Most adults actually like some bitterness if it is not too intense. The third option would be to use something with a very strong flavour to mask everything, e.g. add lots of raw onions and/or garlic, or just make it so hot you don't actually pick up much of a taste. I am not a fan of this solution, but your mileage may vary.
As for a taste combination, which taste should you pick? Strangely, people tend to think that a sweet taste is well suited to somehow "mask" bitterness, which I cannot confirm. For me, both tastes, bitter and sweet, stay there, and one doesn't mask the other, they just create a specifically unharmonious combination. In my experience, bitter + sour is a much better combination (think grapefruit, cranberries, or tonic water) which can even allow for a small amount of sweetness, without making it clash. But it is also good to make a combination of bitter + salty taste. While umami can be combined with bitter too, it doesn't diminish the unpleasant sensation of the too bitter.
As for your specific problem, I often eat basil dips, only I tend to base them on feta cheese. I sometimes add sour cream or other dairy products, but usually less sour cream than cheese. I think that it will be a good solution for your problem, as it will both complement the bitterness with a strong salty taste, and dilute the bitterness concentration as you will be using lots of it. I also almost always add lemon juice (should help for the bitterness too) and a dash of olive oil (which shouldn't affect the taste, only the aroma). Another ingredient which does well in a basil dip are green or blonde olives, if their salt level is not too high.
For a basil dip, I usually throw everything in a bowl and smooth it with the immersion blender. If you aren't sure that you'll like the taste, try tasting a spoon of your current dip with a small piece of feta and/or an olive before committing more ingredients to something you'll ultimately throw out.