A number of answers have addressed the fact that in the United States, the use of gender-specific nouns is becoming less fashionable. However, that does not explain why a female equivalent of senator has never existed, only why if it did exist once, it might be less favoured now.
I try to answer the etymological question here in relation to senatrix.
There is no classical support for senatrix.
The word senator was, both originally in English and in Latin, an expression referring exclusively to males, simply for the unfortunate dearth of female senators. In classical Rome, a senator (Latin sĕnātor) was always a man, and when the word later came to be used to refer to members of foreign legislatures, to my knowledge all such references were to men. Indeed sĕnātor itself essentially meant ‘elderly man’, from senex, ‘old, elderly’.
The obvious female equivalent, sĕnātrix, does appear in Lewis & Short, but a brief search on Perseus suggests it is not attested in any major classical source.
Of course, Rome had two famous female senators in the tenth century mediaeval senate (or what was left of it), and indeed sĕnātrix was the formal title used by Theorda and Marozia, her daughter*. But they are an anomaly, and were I think of little interest when the word senator was imported into English and first popularized.
Therefore we begin with a situation in which anyone who read Latin would know that sĕnātrix was not a classical word, and those of us who delight in imposing arcane Latin grammar rules on unsuspecting English speakers are equally reluctant to pollute the lexicon with artificial Latin words [and anything after August 476 AD doesn't count].
On the use of sénatrice in French.
The French word sometimes used for a female senator (sénatrice) appears to be a backformation from sénateur + -trice rather than derived directly from *sĕnātrix. That said, sénatrice is long established in the French language, appearing in the 1762 edition of the dictionnaire de L'Académie française. This is curious, as in 1762 the ancien régime remained strong and the date long precedes the the establishment of the French Sénat in 1799. However my knowledge of French and France is embarrassingly poor, so all I shall say is that 1762 is long after the word senator appeared in English. Therefore senator had become an English word by this time, and I suggest that the continuing evolution of its cognate in French would have had limited effect on the use of senator in English.
The Académie française now discourages the use of sénatrice.
There is little use of senatrix at any time in English.
A Google Ngram search reveals that senatrix did have a brief period of use in the early twentieth century, around the time that the notion of female senators entered the public discourse. However, a more detailed inspection reveals that this seems to be an upswing in interest about Theodora and Marozia, rather than in the composition of the modern legislature.
In that search I did find one tantalizing reference to the ‘senatrix from Kansas’, but as the reference dates from 1898, eighty years before the first female senator from Kansas, and is entitled ‘Judge's Library: A Monthly Magazine of Fun’, it cannot be taken at face value.
senatrix has little in the way of a history and less in the way of a future.
Lacking a sound etymological past, senatrix was not popular among those most likely to have adopted it; and until 1932 there was regrettably very little need for the word to be used at all in popular discourse.
As the other answers carefully elucidate, in English, particularly American English, there is a trend away from long-established gender-specific words, so it seems quite unlikely we will see the adoption of any artificial ones any time soon.
*You may recall that Gibbon famously describes them in Decline and Fall as ‘sister prostitutes’, but later sources are quite clear that they were mother and daughter. Gibbon does not use the word senatrix as far as I can see.