Can only men become priests or swamis in the Krishna Movement and, if so, why? Why can’t women become swamis?

We should keep in mind that in the Mahabharata the word Acarya is used in a very specific way. The concept of Acaryas, as Prabhupada describes, as the great spiritual teachers who guide the course of Vedic civilization, is a much later use of the word. The title in that sense may have begun with Sankara and was then bestowed on Ramanuja, Madhva, and others.

In Mahabharata, the word ‘acarya‘ indicates a respected professor or teacher and usually a teacher of Dhanur-veda [martial arts and war strategy]. I did a search of the word in the first 300 chapters (Adi and Sabha Parva) of Mahabharata and this is what we find:

1. Drona is regularly called Acarya specifically because he was the official Dhanur-veda teacher of the Pandavas and Kurus.
2. Krpa is also called “best of the Acaryas,” also because he taught Dhanur-Veda.
3. Curiously, even after Krpa taught the Pandavas and Kurus, Bhisma looks around for another teacher and finds Drona.
4. In lists of personalities, the word Acarya by itself always means Drona, even when Krpa is also present.
5. Generally, kings always keep their military teachers, acaryas, near them for consultation and, of course, to honor them at court.
6. Even demigods like Kubera have acaryas in their court.
7. Krishna is called, “father, acarya, guru” at the Rajasuya sacrifice, which shows that Acarya, like father and guru, is a venerable title. Consider how much nonsense the Pandavas tolerated from Dhrtarastra because by dharma, he was their (surrogate) father.

An acarya in the Mahabharata is not a great spiritual leader, as the word indicates much later in Indian history.

In the Bhagavad Gita:

13.12 A component of knowledge is acarya-upasana: serving or honoring an acarya.

In the Srimad Bhagavatam:

5.25.28 Maya Danava is acarya, or “master of the conjurers.”

6.7.7 Brhaspati is the parama-acarya, highest teacher, of the Devas. The term parama-acarya indicates there are lesser acaryas.

6.13.8 The holy name can free one who has killed a brahmana, father, cow, mother or acarya. All these are similarly venerable.

6.19.24 In the Pum-savana rite, one should first seat and feed the acarya before eating.

7.8.2, 8.15.23 Shukra is an acarya (also in Mahabharata)

8.20.21 Shukra is Bali’s kula-acarya, family priest.

9.1.40 King Sudyumna’s royal priest, Vasistha, is called acarya. Vasistha got back the king’s male form by Lord Shiva’s boon.

Thus [also] in the Bhagavatam, acarya does not indicate a great spiritual leader, but rather an important teacher and priest.