Looking at the noun basis on Wiktionary.com, it indicates that the plural is either bases or baseis. It looks like the rare baseis comes from the Greek, but the common bases just refers back to basis and that does not explain the plural etymology. Is there a standard rule at work here, or is this just an exceptional case that arose over time by convention?
-sis, -ses is regular; -sis, -seis is almost nonexistent
Words in Ancient Greek that end in -σις in the singular, -σεις in the plural standardly correspond to words in English that end in -sis (pronounced /sɪs/ "siss" ) in the singular and -ses (pronounced /siːz/ "seez") in the plural. We also see this pluralization pattern with crisis, crises (from Greek κρισις, κρισεις).
Most irregular Latinate or Greek plurals are optional, but for words ending in the suffix -sis such as basis, regular plural forms (e.g. "basises") seem to be quite disfavored (probably due to the awkwardness of having similar sibilants in close succession) to the point that it's probably safe to call them "incorrect" for many words.
"Baseis" is not commonly used
So actually, "baseis" would be exceptional as a plural of "basis." It looks very odd to me; I've never seen it before and my spell-checker thinks it should be "bases". "Bases" is definitely standard, and I would say "basises" is non-standard, but I don't know what "baseis" is. I think it was just included for the sake of completeness, and because Wiktionary can be a bit eclectic. In fact, many of the citations for "baseis" on Wiktionary appear to be from non-native speakers, and others seem to be possibly simple misspellings of either "basis" or "bases." I also can't find any other dictionaries that list it: it's not in the online editions of Merriam Webster, the American Heritage Dictionary, dictionary.com, thefreedictionary.com, or the Oxford English Dictionary. In English, the spelling "baseis" is in no way more correct than "bases." I'd strongly advise against using it.
The only plurals with -eis that seem to be at all common end in -poleis
The Wikipedia article "English words of Greek origin" says
The e form is standard for the plural suffix -εις/-es, following the Latin declension, except in poleis, necropoleis, and acropoleis (though acropolises is by far the most common English plural).
The Oxford English Dictionary confirms that "poleis" is used, "necropoleis" has been used (among various other plurals), and "acropoleis" has been used.
Wiktionary apparently also lists "metropoleis" as a plural form of metropolis (mentioned in Peter Shor's answer to this question about the plural of metropolis). However, in usage "metropoleis" seems to be restricted to scholarly works about Ancient Greece, and the term is frequently italicized, so I would say it's more of a transliteration of the Greek word μητροπόλεις (the plural of μητρόπολις) than a true plural form of the English word metropolis.
-ses is a Latinization of Greek -σεις
According to Wiktionary, the Greek suffix -σις comes from the Proto-Indo-European suffix reconstructed as -tis, plural -teyes. It seems this would have developed in Greek as follows: "y" was dropped between vowels, giving -tis, -tees; in Attic Greek t was assibilated to s before i (this consonant apparently spread to the plural form by analogy) and ee was contracted into the "spurious diphthong" ei, giving -sis, -seis. From Wikipedia: Romanization of Greek
Traditional English renderings of Greek names originated from Roman systems established in antiquity. [...] ⟨ει⟩ and ⟨ου⟩ were simplified to ⟨i⟩ (more rarely—corresponding to an earlier pronunciation—⟨e⟩) and ⟨u⟩.
Besides this plural ending, another example of the e-ει correspondence is the English word panacea, derived from Greek πανακεια through Latin.
One reason why ⟨e⟩ (rather ⟨i⟩) than may have been used for ⟨ει⟩ in the Latin plural of words like this is analogy: Latin already had many native words ending in -is in the singular and -es in the plural, such as axis-axes.
Looking at the different romanization systems laid out side-by-side, you can see that most Greek-derived roots, words and names used in English are spelled and pronounced according to the "Classical" (Latin-based) system. We write "oenophile"/"œnophile" and not "oinophile," "ether"/"aether"/"æther" and not "aither," and so on.
There are some exceptions, mainly words created in modern times, that use either one of the more letter-by-letter transliteration systems for Ancient Greek, or (more rarely) a transcription system based on Modern Greek pronunciation, with equivalences such as η = i, Φ = f.
Additionally, some words exist with both spellings. Sometimes this may cause a change in the pronunciation ("apodictic" /æpoʊˈdɪktɪk/ vs. "apodeictic" /æpoʊˈdaɪktɪk/), and possibly in the meaning as well ("cinematic" vs. "kinematic," "demon" vs. "daimon").
The following answer may also be relevant: How do you decline nouns borrowed from languages with several categories for declining nouns (or none at all)?
Google Ngram shows that "bases" is the normal thing. "baseis" seems to be a dictionary corpse seldomst used.