The explanation most people give is that since the witness has duty to see if the ordinance is performed correctly he should be one who is able to perform the ordinance himself. This is not a very satisfying answer. One doesn't really need priesthood to tell if the correct words were used and correct procedures taken.
Those who say that women should be allowed to witness ordinances usually refer to the New Testament and point out that it was women who were the first witnesses of the resurrection. They also bring forward the fact that YW motto (which is repeated constantly) quotes the responsibility in the baptismal covenant to “stand as a witness at all times, in all things, and in all places”. However, one could claim that witnessing resurrection and standing as a witness at all times is something different than witnessing a priesthood ordinance.
If women were allowed to witness, what harm would it cause? Especially, if young women witnessed baptisms in the temple that would prepare them for other temple ordinances where women have a more active role.
Actually, women have been witnesses in the temple:
The Tribune article tells usIn 1845, for no clear reason, only twenty-four baptisms were registered. The record did record one particularly noteworthy event, however. For the first time a woman, Melissa Lott, was listed as a witness (Nauvoo Baptisms, Book C).
(source: “What Has Become of Our Fathers?” Baptism for the Dead at Nauvoo, by M. Guy Bishop in Dialogue vol. 23, no. 2)
What do you think, should women be allowed to witness ordinances?Until the 1950s, Mormon women routinely served as witnesses in the temple, according to LDS history researcher Ardis Parshall.
[Joseph Fielding] Smith was surprised in 1959, when he heard that women witnessed temple marriages in Alberta, Parshall reports. “He asked for the temple president’s authority, and learned that it was a longstanding practice, based on ‘Item No. 53’ in the written instructions given to temple presidents.”
To that, Smith replied, “Where the idea of having women for witnesses for marriages began I do not know, but my training convinces me that it is the proper thing to have the priesthood not only officiating but witnessing the ceremonies of the temple.”
It has been that way, Parshall said, ever since.
Edited the Tribune quote to tell who Smith is.