On the Greco-Roman side of things: it's not objectionable, so long as you take it for what it is, a reductive condensation of a complicated morass for a general audience. It doesn't get things wrong as such, but it's not representative: it's dominated by a handful of sources, and it divorces the myths from the contexts we know them from.

The handful of sources that it picks are mainstream ones, so there's no dishonesty going on at least. The main sources seem to be Vergil's Aeneid, Ovid's Metamorphoses, the ancient "Lives" of early Greek poets, and the pseudo-Apollodoran Library.* It seems to avoid other sources (e.g. visual arts, Hesiod, tragedy, Orphism, Roman myth, Lykophron, Apollonios' Argonautika, mythographers other than ps-Apollodoros, and late scholastic sources).

It depends what you want to use it for. If it's simply to gain an acquaintance with stories that inspired poets of the modern era, it's absolutely fine -- better than fine, in fact, since it often cites parallels from Tennyson, Keats, Byron, and the like. But if it's to learn about the development of myths in their original context, or to get a picture that's not so dominated by a few texts, it's not so good.

* Edit: note that its use of ps-Apollodoros predates the 1885 discovery of the Vatican epitome, which gave us a précis of the missing end part of the text.

Bulfinch's takes the most common version and elements of the myth and retells them.

The myths in question are not like Grimm's fairy tales (recorded early 1800s) for academic purposes, or even the later Norse myths (recorded 13th cent) which were recorded well after the stories in question ceased to be living stories. There is no single "original recorded version" of the Greek and Roman myths in Bulfinch's. The myths were living stories changed over a thousand or so years by authors in literate civilizations often to suit the story the individual author was trying to tell as well as incorporating local variants into more generally accepted ones. We have variants of the same myths from as far back as Homer (7th-8th cent B.C) to as late as the 10th or 11th century that can all be considered primary sources. We see the myths change and evolve during that time period, so which one is the "original recorded version"?

Remus: a Roman Myth provides a great example. The opening chapters recount every version of the myth involving the foundation of Rome from the 6th cent B.C. to Vergil. Some involved a single founder, some twin brothers, some give different names for the brothers, some involve different parents etc. Its been a while since I read Bulfinch but I'm pretty sure it only covers the standard twin sons of Mars, suckled by a she-wolf who found a city of outcast and Romulus kills his brother Remus. Bulfinch's isn't wrong. It just doesn't give the complete range of stories. The version in Bullfinch's tends to be the most common and popular one. The one recorded by the big name authors and most widely circulated.