While the opinion that the theater was a sinful place, and a generally uncomfortable attitude toward theater practice in general, was certainly around during the years the Globe was in operation, it was by no means a dominant view. In fact, it was not uncommon for acting troupes (such as Shakespeare's) to perform in front of the King or Queen at court. Furthermore, in the years after Shakespeare's death, during the reign of King James, going to the theater became even more popular among the elites. Large, open-air playhouses such as The Globe, gave way to smaller, more intimate (and expensive) theaters meant to appeal to a more affluent audience, such as Blackfriars Theater. The strong anti-theater sentiment you are referring to was much more common among the Puritan faction of Parliament, which, according to Wikipedia, gained control of London in 1642 and banned public theatrical performances. Playacting remained illegal during Oliver Cromwell's Puritan interregnum, but was restored in 1660 with the accession of Charles II.

Basically, theater was not universally considered "vile" and "hellish" during Shakespearean times, and was actually embraced by nobles and monarchs. The Puritans shut down the theaters during and after the English Civil War, but then Charles II (supposedly a writer/poet himself) wholeheartedly endorsed theatrical practices. His court was famed for its theatricality, and theater grew in popularity during his reign.

Sorry for the lack of sources. I majored in English with a special focus on Early Modern/Renaissance theater.

Theatre was wildly popular among all classes of people in Elizabethan England. Let me look at what you're asking from a few angles. First, I'm going to address Elizabethan perceptions of the theatre. The view of its awful, immoral, and overall dirty nature actually comes from mostly Puritan sources. Many Puritans believed theatre to be highly immoral and unChristian, and because these individuals were a part of a rising merchant class of Londoners, they had the money and time to spend spreading this message via pamphlets, preaching, and other public mediums. There is also a sense, from modern movies and books, that the theatres were often closed by authorities because of anti-theatrical sentiments. But in reality, the theatres were closed because of concern for the spread of disease - like the closing of the theatres in 1593 due to an outbreak of the plague in London. There were several theatre venues in Elizabethan London and all of them met with some degree of success and longevity (all were closed in 1642). Thus, the idea that Londoners found theatre reprehensible is, on the whole, very untrue.

Second, let's look at what was going on in the theatres themselves. Yes, it's true that the area nearest the stage on the ground level would have been the cheapest and the most disgusting. There were no public restrooms at these facilities. They covered the floors in walnut shells to soak up the "filth." But, lack of facilities and even things like public urination and general lack of cleanliness was not an uncommon part of most Elizabethan Londoners' lives. In the upper galleries sat wealthier individuals who were able to afford these seats. And it's important to note that they were at the theatre to be seen by the masses as much as to watch the show. In fact, the entire event of going to see a play would have been a raucous social affair. But, people of high influence, like royals, would likely never have attended a public theatre. They didn't need to. Shakespeare himself was a part of the Lord Chamberlain's Men, and later the King's Men, and his company and his plays would have been performed at court so the aristocrats and royals never had need to mix amongst the general London public. This is also the impetus behind the opening of the Blackfriar's Theatre. Wealthier individuals would be able to go to an indoor theatre during the winter and enjoy a social affair closer to what they were used to in their normal social circles.

Were there members of the England's elite that visited the public theatres? It's likely. Many influential gentlemen mention visiting the theatres to see certain plays - Marlowe, Kyd, and Jonson were especially popular. But if these people were already used to traveling through the dirty, smelly, loud, and boisterous London streets, the theatres would have been just another part of that public life.

Lastly, I want to point to what you said about the theatre being against the Protestant leadership. Theatre was certainly political tool in London. So many of Shakespeare's best characters are involved in politics in some way. But, theatre was not necessarily AGAINST Protestant leadership. For one, every play had to be licensed and go through censorship from the Lord Chamberlain's office (the position known as the Master of Revels). What was performed had the official stamp of the government. This doesn't mean that veiled references to anti-Protestant sentiments didn't exist; but I'm going to turn again to the Puritans. Much of what can be labeled as anti-Protestant was aimed at Puritan measures of anti-theatricality and other reforms that were growing as more and more Puritans became politically active. Also, even though Elizabeth and her court were Protestant, they were not so keen on the Puritans. So, it's not necessarily that theatre of the time was anti-Protestant, but anti-Puritan. Something that may have been able to pass censorship and make its way to the stage - further fueling Puritanical anti-theatricalism. Of course all of this builds to the English Civil War, the closing of all theatres in 1642, and the theatre going "underground" during the Interregnum.

Sources: Andrea Stevens, Inventions of the Skin; Oscar Brockett, History of the Theatre, Gail B. Stewart, Life in Elizabethan London; also check out the great documentary by Michael Wood's documentary In Search of Shakespeare