Within the world of the Elizabethan theater, there doesn't seem to have been a social stigma around male actors who played female roles; in fact, it seems like it was understood as a stage in a young actor's education and part of a natural continuum of roles the same actor might play. In accordance with his experience as an artist as well as his physical development, a very young actor might expect to progress from roles with few speaking lines, such as children and servants, to more substantial women's roles, to more substantial men's roles. After that, experienced actors might specialize with roles that suited their own tastes, physical qualities, and skills -- playing venerable older men, for instance, or clown roles that required a more rough-and-tumble physicality and a readiness to improvise. The actors who took on women's roles were apprentice actors, and the reward for their ultimate graduation into full company-member status would likely have been a graduation into adult male roles.

This progress reflects the physical development of a young male actor proceeding parallel to his experience on the stage, with small boys playing small children with few lines, then as youths playing young women as they aged into adolescent bodies and greater experience on stage, and ultimately graduating into male roles that exploited their now-fuller experience as actors and didn't rely on stage conventions to conceal their figures and voices. This trajectory also reflects Elizabethan and Jacobean assumptions about gender and the body, that men's roles were ultimately superior to women's and women's bodies were less perfect or less complete than men's, just as boys were. What would have been said of an actor who chose not to follow the supposedly natural progression from playing boys and women to playing mature men? What would this mean for the career of an actor who found they preferred performing as female characters, for any reason? It would be hard to say. The voices of working actors are relatively absent from sources surrounding the Elizabethan and Jacobean stage, and boy-actors in particular don't get to speak for themselves even when they do appear on the written record. Whether this is a symptom of an insidious kind of stigma within the profession I haven't otherwise accounted for, a sheer lack of written records period, or a consequence of young people being relatively disadvantaged in a field that still ran on money and experience that came with age, I don't know.

The above explanation hinges on the idea that the male actors playing women and girls on the Elizabethan stage were boy actors. When we talk about boy actors, there's not total scholarly unanimity about what kind of "boy" we should be thinking -- should we be picturing a thirteen-year-old playing Cleopatra, or a sixteen-year-old, or a ten-year-old, or a twenty-year-old? Is he only considered a boy relative to the fifty-year-old impresario who pays him, or is he a boy relative to the man-actor he'll presumably one day be? Many writers like Stanley Wells have done complicated math to determine how many boys any given company performing any given Shakespeare play must have been working with to account for the number of women and boy characters onstage in any given scene at any given moment, but I don't know if I've ever encountered a satisfactory accounting of what kind of age span these boy players occupied in an otherwise adult playing-company, where it wasn't a situation where all the roles were being played by boys younger than a certain age. (Which was also a thing, and had its own baggage!) For the purposes of this answer, it doesn't necessarily matter whether we should picture our male actor playing women's roles as a brainy and precocious fourteen-year-old or as a nineteen-year-old with many smaller roles under his belt. Within the career track for actors working contemporary to Shakespeare, including possibly Shakespeare himself, playing leading women's parts does not seem to have been an end career goal in itself for fully adult male actors. Adult men playing comedic female characters may have been a practice, though I'm not sold on that either, but 21st century playing-companies casting men in their 40s and 50s as roles like Rosalind and Juliet, or even Lady Macbeth and Cleopatra, are cleaving no closer in their theatrical practices than playing-companies where adult women play those same roles.

Outside the theater, however, contemporaries weren't so sanguine about men and boys dressing up as women, even in the context of a deliberate suspension of disbelief. Women exhibiting themselves onstage would have been wildly inappropriate to many contemporary moralizers, and males donning the attire of women was no better. The polemicist John Rainoldes wrote in a 1599 pamphlet that "A womans garment beeing put on a man doeth vehemently touch and moue him with the remembrance and imagination of a woman; and the imagination of a thing desirable doth stirr up the desire." (Rainoldes says "a man", but many anti-theater writers specify boys as the agents of on-stage crossdressing; another writer, Stephen Gosson, states that "In Stage Playes for a boy to put one the attyre, the gesture, the passions of a woman", is itself a form of lying.) Even considered apart from contemporary anxieties about sodomy, which were always present around the physical site of the theater as a place people of different ages and classes could meet and mingle, the sight of men and boys wearing women's clothing, counterfeiting women's gestures and activities, was an incitement to lust for the men in the audience watching and getting as horned up as if they were gazing on "real" women. If it didn't diminish their lust that the actual object of desire was male beneath the rich clothes, then audience-goers were on the fast track to sodomy as well as heterosexual lechery. However, it wouldn't have been much better to these anti-theater commentators for an actor to play only men's roles in all-male stage plays -- it was bad enough to play a king when you weren't really a king, or a nobleman when you weren't one, and costumes could be deceptive across categories of age and rank as well as sex. Young male actors playing women's parts do seem to have been regarded with a kind of complex erotic attraction by their audiences and admirers, and as a result to have been seen as somewhat disreputable, but this stigma does not seem to have followed them unchanged into adulthood, even as the stigma around artistic performance did. While the general public up to the highest levels of the nobility enjoyed watching live theater, and may have favored specific actors with something like celebrity status, the occupation itself was still seen as lower in status and disreputable regardless of the roles an actor played.

Some reading:

• "Boy Actors, Female Roles, and Elizabethan Eroticism", Lisa Jardine

• "How Old Were Shakespeare's Boy Actors?", David Kathman

• "‘To serve us in that behalf when our pleasure is to call for them’: performing boys in Renaissance England", Shehzana Mamujee (for more on all-boy playing companies and how an Elizabethan parent may have viewed a child's career prospects in the performing arts)

There's a great paper on the nuts and bolts of learning acting as an apprentice player on the Elizabethan/Jacobean stage that posits more experienced boy actors were often paired with less-experienced boys on stage in order to guide them, but I'm blanking on the author/title -- if I remember I'll come back and edit it in!