Heat the oil in a very large pot. Add the peppers, and allow to sweat for 2 to 3 minutes. Add jalapenos, and salt to taste. Cook over medium-high heat for about 20 minutes, or until the peppers have softened and are partially tender. There will still be some liquid left in the pot.
Add the tomatoes and the basil leaves. Lower the heat and continue to cook for about 30 minutes, until the peppers are tender. Add additional salt, if needed. Remove from the heat.
If canning, add the vinegar, place in canning jars, and process according to the manufacturer's instructions. Or, store in the refrigerator for up to 4 days.
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