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The Knockturn Alley is obviously a place devoted to the dark arts. And everybody knows that. Heck, Borgin & Burkes sold a cursed necklace that could have killed somebody by touching it, and also the Hand of Glory that is really only useful for thieves.
Now then, not much is said about the other shops in the alley, but, as a minimum, why doesn't the Ministry get rid of Borgin & Burkes? Everything in it is really only good for a dark wizard after all.
None (or at least not the majority) of what they openly sell is probably illegal. Dark artifacts aren't illegal in and out of themselves as long as they aren't used to threaten Muggles.
Their patrons include people who have big power with the Ministry (old pureblood nobility like Malfoys)
For that matter, plenty of Ministry holds similar views (Toadface anyone?).
I think the answer lies in why do people get away with a lot of the crime they do now a days. People from the ministry probably support them and keep them in business, like dirty politicians. Remember how Professor Umbridge was not below lying on someone and buying stolen merchandise from a known criminal. I think @DVK's answer tells one part of it, but the other part is probably that they can't be gotten rid of by the ministry is because they are the ministry and they are the wealthy patrons as well.
And also remember that Death Eaters had to be inside of the shop to enter Hogwarts in the 6th book, Some higher ups in the Ministry probably pulled strings so that they had nothing to worry about in terms of resistance.
Borgin probably works with the minstry and sells and gives them dark materials and in turn they let him stay in business.
Here is a qoute from the wikia that to me shows that Borgin and the ministry were friendly.I bolded the parts that I thought were important and canon.
On occasion, Mr. Borgin would purchase items of interest from others; Lucius Malfoy had on numerous occasions sold things to Mr. Borgin to avoid trouble before a Ministry raid on his mansion. In 1992, just before starting his second year at Hogwarts, Harry found himself in the shop, after accidentally exiting the Floo Network one stop beyond his intended destination of Diagon Alley. When Mr. Malfoy entered the shop to sell "some poisons", Harry hid in a Crushing Cabinet. Upon seeing Tom Riddle's Diary, Borgin wanted to buy it because of its "magical qualities". Malfoy, however refrained from selling it, having other plans for it
This qoute and the fact that Borgin once worked with Lord Voldemort when he was younger shows (at least to me) that Borgin and ministry are both corrupt.
And as for your last comment about everything inside of the shop being only good for a dark wizard I would say that that was not exactly true. I think that the Hand of Glory and the Vanishing cabinets could be used by people that are not proficient in the dark arts for "good things".
Also the ministry can't legally banish Borgin and Burkes unless the do something directly to hurt people, which is why they probably were closed or punished after Voldemort's defeat due to the fact they aided in Dumbledore's murder and probably other heinous crimes. Still, Dark arts is not really a punishable thing unless you do something to hurt others if they closed every shop, that did Dark arts they would have to close Durmstrang school and punish Victor Krum, even when doesn't do anything wrong. Its not the Dark arts its what you so with it.
I would consider it similar to how marijuana shops operate in U.S. states where that sort of thing isn't legal- by not explicitly stating that their products are intended to be used illegally, and with no overt criminal activity going on in the shop, Borgin and Burkes are tolerated so long as they don't cross the line, especially as their clientele (i.e. Lucius Malfoy and his cronies) hold positions of some power in the wizarding government.
