There is much uncertainty on details but it seems that some female scholars did have access to at least some parts of the library. It is unlikely, though, that anyone - male or female - could just walk in and browse.
Access for scholars and others
Scholars who enjoyed the patronage of the Ptolemies certainly had access to at least some parts of the library. Unfortunately, there is a lot that we don't know but it is unlikely that just anyone could walk in. Also, the library existed for several centuries so regulations may well have changed over time.
Further, there were different areas of the library (which in turn was part of the Musaeum) with apparently different levels of access. According to the 12th century Byzantine scholar John Tzetzes, at the time of Ptolemy II Philadelphus (died 246 BC),
the external library had 42,800 books; the internal library of the
court and palace had 400,000 mixed books and 90,000 single, unmixed
Source: Robert Barnes, 'Cloistered Bookworms in the
Chicken-Coop of the Muses: The Ancient Library of Alexandria'. Chapter 3 in R. Macleod (ed), 'The Library of Alexandria' (2004)
According to Barnes, based on the information we have from Tzetzes, it appears that the larger, internal library was only available to scholars of the Musaeum.
We tend to assume that the Great Library, from early on, must have
contained the important, canonical works of Greek philosophy and
medicine, and that local and visiting scholars must have had ready
access to them. Circumstantial evidence certainly supports this view.
It seems that some scholars enjoyed patronage, tax free status, the
right of residence in the royal quarter of Alexandria and that the
Library and Museum were not linked to any particular philosophical
school or doctrine....Groups of
scholars from the late fourth century BC onwards, were able to take
meals together in the Museion
Source: John Vallance, 'Doctors in the Library: The Strange Tale of Apollonius the Bookworm and Other Stories'. Chapter 5 in R. Macleod (ed), 'The Library of Alexandria' (2004)
These scholars, enjoying the patronage of the Ptolemies, were probably approved by the Head Librarian who was himself (no female head librarians apparently) appointed by the Pharaoh. Among them (at different times) were Archimedes, Callimachus, Euclid, Herophilus, Hipparchus and many more.
Women at the Library of Alexandria
Women were not among the initial scholars of the library. In the early days,
...encouraged by Ptolemy II (Philadelphus), as early as 283 BC there
came together what Strabo later called a synodos (community) of
perhaps 30-50 learned men (there were no women), salaried members of a
'civil list' for their services as tutors, granted exemption from
taxes, and given free board and lodging in the royal quarter of the
city, where in a circular-domed dining hall, they communally dined.
Source: Roy MacLeod, 'Introduction: Alexandria in History and Myth'. In R. Macleod (ed), 'The Library of Alexandria' (2004)
According to Sarah B. Pomeroy in Women in Hellenistic Egypt: From Alexander to Cleopatra, women were not favoured with scholastic financial support in the way male scholars were by the Ptolemies. If correct, this avenue of access may have been very limited to them. However, Cara Minardi, in Re-Membering Ancient Women: Hypatia of
Alexandria and her Communities, citing Jane Rowlandson in Women and Society in Greek and Roman Egypt: A Sourcebook states that
...some of the international scholars who worked in the Museum were
women (Rowlandson 303-304).
Two such examples cited are the grammarian Hestiaea and the poetess Diphila. Whether or not scholars had patronage probably did not on its own mean that one was denied access to all parts of the library.
One means of access would probably have been having the right contacts (so social class would have been important, as llywrch points out in a comment below). Many daughters followed in their father's footsteps in the liberal arts (see the daughters of Diodorus Cronus, for example); thus, if the father had the patronage of the Ptolemies and thus access to the libraries, it is not inconceivable that a daughter who had a reputation in her own right might have been admitted to the external library; there were certainly female scholars in Alexandria, and female literacy was not uncommon among Greek Egyptians.
On a final note, and contrary to what one might assume given the historical period, female scholarship can be dated back thousands of years. Pomeroy names several female scholars from the late classical or early Hellenistic periods, including Nossis, Anyte and Hedyle, although none are of these are known to have visited Alexandria. Also, Plato had female students (e.g. Lastheneia of Mantinea and Axiothea of Phlius), and there were Pythagorean women philosophers in the 6th and 5th centuries BC.
The point here is that the idea of female scholarship would not have been alien to those deciding who was granted access to the library, so we should not categorically rule out the possibility that women could access the library. At the same time, though, it is also true that female scholars were frequently belittled and discriminated against.