It is permissible to do work on Rosh Chodesh. Ideally, though, one should not work on Rosh Chodesh, similar to the law regarding Chol HaMo’ed. The holier the day, the more it is designated for spiritual endeavors, and the more one should refrain from working. Thus, all work is forbidden on Shabbat, because it is the holiest day. Work is also forbidden on Yom Tov, a level below Shabbat, but one is permitted to do things necessary for the preparation of food. Chol HaMo’ed is the following level, when only certain acts are forbidden. Rosh Chodesh should rightfully be on par with Chol HaMo’ed, but since the Twelve Tribes sinned in the episode of the Golden Calf, Israel lost, correspondingly, the special quality of the twelve Roshei Chodeshim of the year. Women, however, did not participate in the sin of the Calf, refusing to contribute their nose-rings towards its creation. Therefore, God gave them reward in this world – “They keep Rosh Chodesh more than the men do” – and in the World to Come, for they will eventually regain their youthfulness, like the moon which renews itself every month, as it says, “Who satisfies your ornament with goodness, so that your youth is renewed like the eagle” (Tehillim 103:5. SeePirkei DeRebbe Eliezer 45). As a result, women experience more of the sanctity of Rosh Chodesh; and they have adopted the custom not to do work on the day. .
When the Temple stood and the kohanim offered Musaf sacrifices, some men also had the custom to refrain from doing major work on Rosh Chodesh. But their custom has no binding force, because men committed the sin of the Calf. Women, on the other hand, who did not sin, have more of a connection to the sanctity of Rosh Chodesh and their custom has validity.
Therefore, every woman must refrain from doing some type of work on Rosh Chodesh – knitting, for example – in order to differentiate between Rosh Chodesh and an ordinary day. It goes without saying that she may not plan to do any big jobs on Rosh Chodesh.
Meticulous women refrain from doing any type of work that is forbidden on Chol HaMo’ed. This includes sewing, knitting, and fixing things in the house. Cooking, baking, and ironing, however, are permitted, as they are on Chol HaMo’ed. Washing clothing in a washing machine is also permitted, because it entails almost no effort. If the clothes are needed for Rosh Chodesh itself, one may wash them even by hand. A woman who works for a living is permitted to do so on Rosh Chodesh, even if she generally adopts the customs of the meticulous, for if she misses work regularly on Rosh Chodesh, she might lose her job. And even if there is no concern of being fired, she may go to work if she needs the money, or if her absence will cause damage to her employer.
 This is how Pirkei DeRebbe Eliezer (45), the Tur (Orach Chayim 417), the Perishah (ibid. 1), and the Darkei Moshe (ibid. 1), quoting the Or Zaru’a, explain the issue. See Sefer Rosh Chodesh, beginning of ch. 11, where the author recounts all the reasons given for this matter. R. Chayim Vital explains in Sha’ar HaKavanot (76b) that women correspond to [the sefirah of] Malchut (Kingship) and the moon, which renew themselves, while men correspond to Tiferet (Splendor), which does not. Furthermore, Malchut undergoes a decline, from which it can reach even higher than Tiferet. A hint for this is “A virtuous woman is the crown of her husband” (Mishle, 12:4).
According to the Shibolei HaLeket, when Rosh Chodesh is two days long, the custom for women not to work applies only on the second day, which is the first of the new month. The Roke’ach holds that it applies both days. The author of Mor U’Ketziah maintains that the custom takes effect only during the day of Rosh Chodesh, not at night. It seems to me that women keep this custom at night, as well. See Mishna Berura (417:4), Bi’ur Halachah (end of 417), and Sefer Rosh Chodesh (11:8-9).
The Bi’ur Halachah (417, s.v. minhag tov) writes in the name of the Bach that a husband may not demand that his wife work on Rosh Chodesh, but she is permitted to do work if she pleases. As stated above, most poskim agree that a woman should, in any event, refrain from some form of work. Clearly, since a woman has a mitzvah to refrain from work, even though it is not obligatory, her husband cannot demand that she do work on Rosh Chodesh, except housework, like cooking.
The Chida, in Yosef Ometz (20), cites Rishonim who hold that men, as well, had a custom not to work on Rosh Chodesh, and he explains that this custom developed because they used to bow before God in the Holy Temple. The Turei Even (on Tractate Megillah) claims that the custom was based on the musaf sacrifices that were offered in the Temple on Rosh Chodesh. Either way, most poskim believe that this custom has no binding force, as the Mishna Berura writes in 417:2.