The Halacha established that it is permissible to wear regular (as opposed to reading) eyeglasses in the public domain, without an Erub, on Shabbat. However, many people have a tendency to remove their glasses when it is raining, in order to dry them. Therefore, one must be cautious not to carry them while wiping them. Rather, he should stop and stand in one place until he puts the glasses back on.
Rav Shlomo Zalman Auerbach (Jerusalem, 1910-1995), in his MInhat Shlomo (2:35) writes a Chidush that one is not allowed to wear eyeglasses in the public domain when it’s raining. Clearly, this is not the prevailing practice. Even his student, the Shemirat Shabbat K’hilhata (18:48) found room to be lenient. The Mehkereh Eres was also lenient, as long as one is careful not to remove them while walking.
SUMMARY
It is permitted to wear eyeglasses in the public domain while it is raining, as long as one is careful not to remove them while walking.