Why shouldn't the government keep tabs on a group whose professed ideology is the elimination of government?
But I don't think that, in a free country, the government should be monitoring people solely for their political views or what kind of books they read, without some actual evidence that they're a danger to anyone. Sure, maybe someone who buys an Ayn Rand book will eventually work to eliminate government, but I'm not sure owning The Fountainhead should land you on a watchlist; and the same should go for reading Proudhon or Kropotkin.
It's also really easy to run into false positives. In the '80s/'90s, for example, the police/media liked to paint a bunch of generally harmless BBSing kids as "dangerous anarchists" because they had an ASCII file of The Anarchist Cookbook—which is violent anarchist literature, after all.
Curiously, communists probably had it better precisely because of how focused they were on the one "big" goal (communist revolution): Leninist parties in particular were very centralized, disciplined, and scrupulously avoided doing minor illegal things (vandalism, etc.) that could give the police a excuse, because Marxist-Leninist ideology generally believes that small-scale direct action is ineffective, and the vanguard party should keep its powder dry, so to speak, until they're in position to seize power in one quick move. Anarchists are much more decentralized, and have varying opinions about how to organize change, so there's much more scope for the cops to find someone who spray-painted or vandalized something, and via that argue that any group that person was part of is part of an "anarchist criminal cell" by transitivity (occasionally they get tired of waiting, and a police infiltrator will actively egg on more militancy).
What you are proposing is that the government spy on anyone who falls within a broad category of political classification. I think the issue with that should be self-evident.
"Our aim is not to overthrow the state, but to ignore it. Anyone who wants to continue to support the state and obey its laws is free to do so, so long as they leave us alone. Our goal is to build the kind of society we want, and prevent the state from overthrowing us while we’re doing it. The last person out of the state can turn off the lights." - Kevin Carson
To some anarchism is the idea of a completely voluntary society. A violent overthrow of the state is against this idea.
Then they came for the socialists, and I didn't speak out because I wasn't a socialist.
Then they came for the trade unionists, and I didn't speak out because I wasn't a trade unionist.
Then they came for me, and there was no one left to speak for me.
Aside from that general principle, there's also the reality of how its implemented. Most of the time sleeper agents are used to get evidence for trumped up, meaningless charges like "interstate travel with intent to riot." That's the best case scenario. At worst, these agents, on not finding something of even minor note, will attempt to incite violence to prove that "those anarchists have really been violent all along." See, for instance, Brandon Darby.
Edited to add: I do, for the record, think that if there's actual evidence that someone's trying to blow up a bridge, monitoring is appropriate (though I don't see why simply arresting them wouldn't usually be sufficient if that were the case). But usually these things are more fishing expeditions than anything, against people whose ideology the State finds distasteful.
Be careful, this is only true in certain countries. Sometimes governments are created by someone who is the best warlord & administrator, c.f., Charlemagne.
If the people present a democratic threat to the government, the government should fade from existence peacefully. If people present a non-democratic threat to the government, then the government should act to protect itself.
Anarchists do not, categorically, present (or seek to present) a non-democratic threat to the democracy.
Do you want to keep tabs on violent* anarchists? Knock yourself out. I have no particular issue with that.
I am an anarchist (a market anarchist, specifically). It is my professed ideology. I don't break laws (well, maybe occasional acts of non-violent civil disobedience in protest of specific unjust laws), I pay my taxes, and I behave according to pretty high ethical standards.
I wouldn't be surprised if there are files on me with the Austin Police Department, and DHS (via the Fusion Center here in Austin), because an organization I've been a part of has been infiltrated multiple times by undercover APD officers (and APD has, thankfully, been slapped around by judges for doing so, and for the way the officers behaved in the process). But, I'm not pleased about it, and I consider it a violation of basic civil liberties provided by the constitution...since I've never been arrested and never been charged with a crime. I am, by all reasonable standards, an upstanding citizen. Only if you believe holding a belief can be a crime, could you believe that I deserve to be spied on.
I'm sure you hold some beliefs that others, including others in positions of power, consider wrong. Wanting to change (or dismantle peacefully) the government is not a crime.
In short, it aint nobody's business what my ideology is. And, if you believe someone should be put in prison, in solitary, for holding an unpopular ideology...well, that's just not very American. Worse, it's not humanitarian.
Hmmm - yes, why not: http://www.gop.com/2012-republican-platform_Reforming/#Item4
Is it the very word "anarchist" that causes the difficulty, because that's what it looks like to me.
It's of course reasonable to keep an eye on seditious groups: these almost by definition imperil life, liberty, property, etc. It is another thing altogether to keep an eye on someone who wants to peacefully change government via peaceful and lawful means.