To the outside world, the SS was the upper-class, elite institution of the Nazi party. Historical accounts describe the SS as attracting a "better sort of people" and as being the "most respectable of the Party organizations." (That last quote is by the Archbishop of Freiburg, by the way.) Many Germans saw the Weimar Republic as evidence of what happens when the natural elite is not recognized and given their natural place of power, and so the idea of an institutionalized organization of "superior" people excited feelings of envy and admiration rather than derision.
This was especially true because Himmler was a master at presenting this image of the SS in romantic terms near and dear to the German heart. The uniform of the SS officer is either admired or ironically mocked, but keep in mind that it was expressly designed to instill fear and admiration into the viewer --- and it was very effective. Entirely black and silver, it also incorporated various occult symbols into a complex display of rank and status.
Ironically for an institution founded on the corruption of big business, Himmler flung the doors of the SS wide for the German upper classes. He needed money and prestige for his fledgling institution, and those could only be had by from the old money, the businessmen, and the noble families. By 1933, almost one in five SS-Obergruppenfuhrer (generals) of the SS had the "von-" prefix to their name. (The prefix indicates noble status.) For Brigadefuhrers (mj-gen), this was about one in seven. For Standartedfuhrers (colonels), about one in twelve. For the general population, one in 134. The rolls of the SS included Dukes, several Counts, and a Prince. The rank of honorary commander (which is what Klaus is) was bestowed upon high-level industrialists, diplomats, scientists, and Party and government officials (some of whom wanted to kill Himmler personally). Many of the Equestrian Societies (important country-club like organizations for the rural elite) were incorporated into the SS wholesale.
As opposed to this elite leadership, the rest of the SS came from the middle and lower classes. The administrators and middle managers of the SS came from the universities, sons of the middle class. Many joined for status or to avoid the combat positions of the SA. Most had university educations. This class of recruit might be an administrator in one of the various departments, or high in the management of one of the WHVA industries. The men of the SS, the grunts, came from the farming classes, peasants who could no longer make a living off the land. These recruits usually had no education whatsoever. This class usually found itself in a Waffen-SS unit or a concentration camp guard (though the bright could enroll in SS-cadet academies and become officers.)
Himmler's publicity campaign worked. Dozens of bankers and industrialists joined "Friends of the SS" committees and donated money in exchange for honorary memberships. (Also as a way of bribing the Nazi Party. The SS was seen as the go-to organization if you wanted the Nazi Party to -not- confiscate or commandeer your industry for the war effort.) Members of these committees included the Chairman of IG Farben, the President of the Reichbank, and the Chairman of Vereinigte Stahlwerke. Millions of RM were poured into a special Reichbank account. Less wealthy applicants could become "sponsoring" members of the SS. In 1934, there were over 342,000 such "sponsors" contributing 581,000RM.
The image of the SS as noble elite extended beyond mere dress. To give one example, the SS revived the long-forgotten practice of duels. Actually, this example is instructive: While Heydrich (the second-in-command after Himmler) was winning fencing tournements, Himmler was drafting the regulations for duels of honor. Heydrich was everything Himmler was not: tall, fair, handsome, charming, and charismatic. Himmler was actually physically unremarkable, with a soft face hidden behind glasses. It is easy to underestimate Himmler, however. While Heydrich brought verve to the part and the enthusiasm that gave birth to the Final Solution, it was Himmler that turned a 300-man brute squad into a million-man network of terror and oppression. Those who dismissed Himmler as a mere accountant or bureaucrat often were taught differently with a bullet. And he managed to do this while maintaining in the public eye the image of the SS as the new chivalrous order of knighthood.
Himmler created three signs of favor within the SS. The lowest was a death's-head ring. The next was a ceremonial dagger, usually awarded only to officers. The last was a coat of arms, drawn up by the Abnernerbe specifically for the so-favored. Himmler considered these twelve men to be his knights, and had plans to create a new `round table' with them. Over the course of the war, he spent over 13,000,000RM to have a castle restored for his use. He drew up plans for occult rituals to be help there wherein he and his knights would meditate and commune with the dead. Fantastic ceremonies were sketched out for their eventual cremation and burial in the castle's lowest chambers.
Himmler believed in communing with the past leaders of Germany, and believed himself to be able to call them up at will. He thought himself the reincarnation or heir of Heinrich I (835-936), and believed that Heinrich I spoke to him when he was half asleep. He founded the "Society for Preservation and Promotion of German Cultural Monuments" (which did just that) and the Abnernerbe. He created new yule-tide, solstice, marriage and birth ceremonies for the new Germano-pagan state. Christian priests were to be expelled from society.
The SS was consciously and intensionally patterned after the Jesuits. Fanatical devotion to Hitler replaced fanatical devotion to the Pope, and the Reichfuhrer-SS replaced the Father Superior. Both organizations granted fantastic privileges to thier members, were above temporal/secular jurisdiction, and instilled blind obedience to the leader (Pope/Fuhrer). Like the Jesuits, the SS required their applicants to undergo extensive and grueling training before being allowed to take the "kith and kin" oaths of membership. (Unlike the Jesuits, the SS applied racial filtering based on physical appearance.) There was even a "catechism" they needed to learn.
There were two things that Himmler would not tolerate in his troops: a drop of Jewish blood, and homosexuality. As Colin mentioned, all SS members were required to prove an ancestry free of Jews for themselves and their wives back to 1800 (1750 for officers). Any member who found a drop of Jewish blood in their veins had to report it and request release. If found out, they were expelled. (If in the lower ranks; this was occasionally waived for the generals.) In some cases, marriages were forbidden when the RuSHA found Jewish ancestors in the bride's family tree.
Even long time, high-level members were expelled over homosexual encounters, and in 1937 Himmler signed an order instructing homosexuals in the SS to be turned over to the courts. Considering that the SS was officially above the law and a state unto itself, Himmler was instructing that homosexuals were retroactively never members of the SS. Also, after serving their prison sentence, such never-members of the SS were to be "sent on my instructions to a concentration camp and shot while attempting to escape."
For the record, I am making none of this up.