Can languages with infinite strings be recursively enumerable?
As you state, the standard definition of TM acceptance requires that the TM halts after finite time.

It is still possible for a TM to accept languages that contain infinite words, if they can accept without reading the entire word. For example the language of all (finite and infinite) words that contain at least one $0$ is semi-decidable, since if there is a $0$, the TM will encounter it after a finite amount of time and can then accept the word.
The complement of this language however is not semi-decidable, since it is not possible to read an infinite input in finite time (and it is not possible, in the general case, to know that there is no $0$, without reading the whole input).