The main aspect of any card in dominion is whether it is terminal or not; Lost Arts completely changes that balance. Moat with Lost Arts becomes better than lab, while costing only $2. This happens for many cards, and it isn't hard to think of more examples of cards that become very, very strong as a result of no longer being terminal.
The question, then, is where and when do you buy Lost Arts, and what do you put it on?
It goes without saying that Lost Arts is an engine card; getting +1 action on a card that you will usually only play once a turn isn't a great help.
Anyway, there are two main options in regards to Lost Arts: you can either put it on your terminal draw (think smithy or moat) or put it on your terminal payload (think bridge or goons or mountebank). Both are good options. Obviously, both aren't going to be available on every board that lost arts appears, either. Anyway, lets look at where each approach is good and where each approach fails.
The terminal draw option is quite strong. When there aren't any villages, it can become almost mandatory; even when there are villages, you can simply either ignore the villages or buy a few at the beginning then later stop buying them when you get lost arts onto your terminal draw card, depending on what other terminals you want to play. This somewhat depends on which draw card you are using; cheaper ($4-?) draw cards will like having villages more while more expensive ($5+?) draw cards like getting villages less. This is because the gap of time, and thereby the utility of villages, of smithy to lost arts is longer than that of, say wharf to Lost Arts. Even in the latter case, villages may still be useful if there are some strong terminal payload on the board that you would like to play multiple of. It should also be noted that putting +1 action onto a terminal draw makes your engine much more reliable, as you no longer have to worry too much about drawing your draw without any villages to play it. Finally, just as there not being any villages helps putting your +action onto your terminal draw, having multiple villages also helps because you are't running around trying to fight over the village pile. This approach becomes weaker if there are nonterminal draw cards already in place, especially without villages, as sometimes it may be simply better to buy a lab and put your +action onto a terminal payload rather than worrying about getting lost arts on said terminal draw then getting more villages to play your terminal payload.
The other option is putting it on your terminal payload. This is actually much, much stronger than it may sounds at first, to the point that when it is available, I think that it is usually stronger than putting it on your terminal draw, although you obviously only want to do so when there is a good, spammable terminal payload. Consider first that engines are often able to consistently draw themselves without Lost Arts on a draw card. The next thing to consider is that you can, potentially, get far, far more use out of lost arts on a terminal payload card than a terminal draw. For example, a deck with 7 bridges may only need something like 4 smithy's; putting +1 action on your smithy versus bridge gives 3 less actions over your turn, whereas putting the Lost Arts onto the smithy doesn't do much to increase payload capacity, only reliability. Lost Arts on terminal payload also works wonders when your draw is a draw-to-X card, as your terminal payload is now almost a cantrip, and you no longer have to worry about getting enough villages to play your terminal actions before you play your your draw card either, because before you needed a +action for every terminal payload you want to play. Putting Lost Arts on your payload also works better when villages are contested, as the number of villages needed to play out all the payload you need is usually much larger than the number needed to play all your draw cards. Further, putting Lost Arts on your payload works much better in the general timeline of how your deck is likely to develop anyway because you never have those awkward questions of whether to buy a village or overbuy draw cards.
As usual with dominion, there are some womewhat edge-casey stuff to consider.
An option is to put lost arts on a card that is already nonterminal to create a village when there aren't any on the board. This is uncommon, but sometimes useful when the you need to play multiples of both your terminal payload and terminal draw, or if there are multiple different terminal payload cards that need playing.
In games with 3 or more people, it can often be good to place your +action onto a different card than other players to avoid competing with them. I can't say too much about this, as I don't really play 3+ player dominion that much.
In summary, basically Lost Arts works well when put on terminal draw, but usually works better when put on terminal payload. But don't be an idiot about it either; Lost Arts on Wharf is probably better than Lost Arts on woodcutter.
