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It appears that a left Bousfield localization of a cartesian model category is again cartesian. This can be deduced fairly easily from Lemma 7.7.3 in Simpson’s book, which proves that weak equivalences in a cartesian model category are closed under products.
Is there an explicit reference for this statement anywhere in the literature?
That can’t be right. Dugger’s theorem says that every combinatorial model category is Quillen–equivalent to a left Bousfield localisation of (the Heller model structure on) a simplicial presheaf category, and those are always cartesian. But being a cartesian model category implies that the homotopy category is cartesian closed, and this is invariant under Quillen equivalences; it should go without saying that there are combinatorial model categories whose homotopy category is not cartesian closed.
Indeed. Charles Rezk in Section 2.21 of his paper “A cartesian presentation of weak n-categories” explicitly points out the need for a cartesian saturation to ensure that the left Bousfield localization is cartesian.
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