There isn’t any semantic difference between “let him be cursed” and “a curse be on him” and “he is to be (ac)cursed”.
“That person is cursed” is more neutral, and might be only a statement of fact, rather than something Paul agrees with or enacts to whatever degree – but then again in the same 1 Cor, Paul hands the Stepmom-Sleeping Guy over to Satan for the destruction of his flesh so that his spirit may be saved in the Day of the Lord to come. Regardless of what “destruction of the flesh means” (whether to death or not), Paul is actively inflicting the judgment or at least agreeing with it strongly.
Anathema literally means up-place, and metaphorically means being held up as an example, by cultural context as a bad example or as an example of what happens for bad behavior. Knoch thinks it relates back to pagan human sacrifices originally as punishment! 
In the Greek LXX however it has a much more neutral meaning of being devoted to God, sometimes devoted for destruction where appropriate.
Anathema is itself a noun; there’s a verb form (not the verb from which the noun comes), that’s rarely used in the NT, most importantly when Peter curses himself to deny Christ in Mark 14:17 (so an anathema isn’t hopeless. 
The other verb usage is toward the end of Acts when some Jews put themselves under an oath to neither eat nor drink until they have killed Paul. This presumably does not work out well for them. 
Most of the time the word is used as a noun, which has six occurrences in the NT (one of which is the aforementioned Acts 23 assassination conspiracy.) Aside from the examples already mentioned in Galatians 1 and 1 Cor 16, Paul could wish himself anathema from Christ if that would save his fellow Jews in Romans 9:3, and again at 1 Cor 12:3 Paul writes that no one can say “Christ is anathema” while speaking by the Spirit of God.
The noun form is always the same, so translations into English are taking the noun and turning it into a verb, in relation to whatever nearby verb of being is contextually connected to it. For 1 Cor 16:22 and Gal 1:8-9 (where Paul repeats the curse twice for slightly different reasons), that verb is {êtô}, which is the 3rd Person Singular form (providing the implied “him”, by the way, which doesn’t show up in Greek there) of the present imperative active form of “is”.
In other words, “let him be cursed” and “a curse be on him” and “he is to be (ac)cursed” are all proper translations. “That person is cursed” doesn’t translate the concept of present active command or requirement. It’s the difference between saying “let him be stopped” and “he is stopped”; the former is the correct parallel to the verb form, not the latter.
There is also one NT usage of a slightly different noun form {anathêma} to refer to votive offerings in the Temple, as Luke translates the disciples calling attention how beautiful the Temple looks in GosLuke 21:5.
Anathema is actually a noun form of the verb {anatithêmi}, which is itself a prolonged version of the verb {theô} from which {theos} comes – but it doesn’t necessarily have anything to do with God. The underlying idea is to rise or more basically to place, and so by application there’s a sky/solar connection, in which the sun rises and from where generation comes (it’s also the primary word from which, more directly but much later, we get the English word “day”) and also a thematic connection (not grammatical) to {logos} by the way. The verb form with a slight modification into {theôreô} can also mean to look upon, which is probably a derivation from the idea of the sun (seeming to be) looking down from on high. You might recognize the {-tith-} part as being where we also get the word “tithe”, which goes back to the more usual LXX application of devoting or sacrificing a gift to God.
The verb {anatithêmi} shows up twice in the Gospels, both times in relation to Paul as it happens. By context each time, it means placing-up Paul’s case before someone, such as in his legal defense to Festus, or Paul submitting for examination, to the Jerusalem apostles, the gospel he received from Christ.
I had a vague theory, but it didn’t pan out – I checked a grammatic interlinear and both forms are listed as “vm Pres vxx 3 Sg”. (I don’t understand several of the verb form abbreviations in that interlinear, including vxx in this one, or the m after the v – it isn’t masculine because sometimes the letter is ‘i’ or something else like that. 
Any ideas…? I feel like the initial letter possibilities after “v” for verb, dug, are something I ought to recognize… 
But ‘i’ sure doesn’t stand for intransitive, I think…?)