There are two complications here. First, the idiom "based on" in English doesn't have a single meaning. It is used in a variety of situations with somewhat different meanings; in the sentence "Shakespeare's Comedy of Errors is based on Plautus' Menaechmi" the phrase "based on" is not doing the same work as it is in the sentence "we punish criminal offenders differently based on the results of their actions." So in order to answer your question accurately, we first need to identify exactly which sense of the idiom "based on" you are trying to find a matching idiom for in Latin.
I'm guessing, however, (based on the concrete example you provided) that you want to find something more like "Shakespeare's Comedy of Errors is based on Plautus." Now it is unlikely that the image contained in the English idiom (that is, of creating a "foundation" or "base" and then putting something new on top of it) is going to be the exact same image used in Latin to express a similar thought (and corpus searches confirm this). So in order to translate the phrase into idiomatic Latin, we will have to find passages in the Latin corpus that express the same general idea but that use different words and formulas.
To reiterate: If we want to do more than merely dress up an English, Italian, etc. image in Latin clothing (which, with respect, is what "basatus est" would yield), we have to figure out not how to translate the words but the idea itself into idiomatic Latin.
Prologues in Terence will be extremely helpful for finding an idiomatic way of expressing the idea that plot material or the entire plot of one work is drawn from another, since most of the plays are based on earlier Greek versions and the prologus often makes reference to this fact:
Heauton: "ex integra Graeca integram comoediam
hodie sum acturus H[e]auton timorumenon, 5
duplex quae ex argumento facta est simplici."
Andria: "Menander fecit Andriam et Perinthiam.
qui utramvis recte norit ambas noverit: 10
non ita dissimili sunt argumento, [s]et tamen
dissimili oratione sunt factae ac stilo.
quae convenere in Andriam ex Perinthia
fatetur transtulisse atque usum pro suis."
Adelphoe: "Synapothnescontes Diphili comoediast:
<³ea>³m Commorientis Plautu' fecit fabulam.
in Graeca adulescens est qui lenoni eripit
meretricem in prima fabula: <³eu>³m Plautus locum
reliquit integrum, <³eu>³m hic locum sumpsit sibi 10
in Adelphos, verbum de verbo expressum extulit. "
Eunuchus: "qui bene vortendo et easdem scribendo male
ex Graecis bonĭs Latinas fecit non bonas,"
So what we see from the above examples is that there might not be a single formulaic phrase in Latin that perfectly matches the specific sense you are looking for. Rather, there are several phrases which can be used to describe the more concrete (and more general) idea of taking something from one thing and using it to make something new. So the sentence "Shakespeare's Comedy of Errors is based on Plautus" might be variously rendered as "Shakespeare ex fabula Plautina Comediam fecit suam Errorum," or "Shakespeare argumentum ex fabula Plautina sumpsit sibi in Comediam Errorum," or perhaps "Shakespeare quae convenerunt in Comediam Errorum transtulit ex fabula Plautina atque usus est pro suis."