I tried giving basil once, but only as a treat. At first they weren't sure if they could eat it. They would just sniff and hop away. But I would just leave it their bowl, few moments later, they'd eat it!
I haven't tried basil or other fresh cooking herbs. I have a great variety of forage to offer Dosiedoe. Her favorites are dandelion and Jerusalem artichoke. Plaintain (plantago major and p. virginia) are another favorite.
Basil has a carcinogen as one of the components of its odor and flavor. Doses of that chemical less than 10mg/kg rabbit present little risk. Given that that chemical is a very small part of basil you would need to feed bales of basil over a long period of time. Otherwise it has some carbs and protein to offer, but terrible rich. It is not loaded with sugars. I would say it isn't bad for rabbits but there are better forages to be had.
Basil (and many other herbs) are fine for generous daily feedings. I grow it in an herb garden each year just for my rabbits! There are some greens, like parsley and spinach, that should be limited or rotated. High oxalates or high calcium is usually the reason. You can see a sample list of which greens are safe for daily feeding and which should be limited here: