In her song “Language Is a Virus,” Laurie Anderson says “paradise is exactly like where you are right now, only much much better.” Well, traipsing is exactly like walking, only much much worse. “I went down to the store”: neutral. “I walked over to the store”: neutral. “I traipsed over to the store”: you hated every step.
Why was it bad? Why is traipsing always bad? Sometimes it’s because you’re going through mud or snow – walking dirty, you might say. Sometimes it’s because you’re tired or it’s a tiring trip, perhaps needlessly so. Sometimes it’s because you’re bored. Sometimes it’s because you have to and you don’t want to. But traipsing can’t be positively toned.
It’s not to say that the destination or cause of traipsing is necessarily undesirable. Heck, that’s often the only reason you’re traipsing at all! “I sure am glad to see you! I traipsed through twenty blocks of snow drifts to get here!” Or perhaps “I traipsed all over hell’s half acre that summer with the most beautiful person I had ever seen.” But if you say “It’s been nice traipsing around the city with you,” you are assuming that the other person agrees that it has at least been a considerable physical effort – or understands that you’re being playfully ironic.
The word itself seems to take extra effort in getting to its destination. Why not a shorter spelling, such as trapse or trapes? Admittedly, there is the issue that trapsing looks like the a is short, and trapesing looks like the e might not be silent, but in fact both spellings have been used in times past. Not only that, it has also been rendered in the 1800s as trapess, trapus, traipass, and some similar words, and pronounced accordingly.
Which kinda suggests an etymology, doesn’t it? Indeed, the source of traipse is thought by many to have been trespass, or, more to the point, its more recent French version, trépass. Not everyone agrees, though; there is an old word trape that seems to trace to Middle Dutch and Middle Low German trappen, ‘tread, trample’. But there’s something of a gap between that word and this one. Sorting out with certainty which is the real source will require more legwork.
What we do know is that traipse has that tr- onset that shows up in some other words relating to effort: trudge, tramp, travail, try, and trek, to name a few. It also has that “long a” (/eɪ/) for extra effort, plus a final s that is not a plural. But of course none of that has any necessary bearing on the sense.
Well, neither does the fact that it’s an anagram of parties, which are quite the opposite of traipsing and yet at the same time are good motivations for traipsing through bad weather. But the relation of sign and signified is supposed to be arbitrary, however much fun we may have finding extra ways to enjoy it. On the other hand, we often focus too much on the denotation and not nearly enough on the connotation, as though the subtle sense differentiations were just something to get through to reach the goal of dictionary meaning. But the trip is worth it, I think, if only for being able to tell of it after.
Sesquiotica 
Not everyone agrees with my view on the sense of this word, I find! But I was basing my characterization not just on my own experience and sense of it but on the OED’s definition, “To walk in a trailing or untidy way; e.g. to walk or ‘trail’ through the mud; to walk with the dress trailing or bedraggled; to walk about aimlessly or needlessly. (Usually said of a woman or child.) Also in gen. use, to tramp or trudge, to go about.” In every instance of use I have seen, there is a tone that is either necessarily or possibly negative. But it’s fun to see some debate on the subject! (I wasn’t trolling, though.)
With regard to the question of tr-, by the way, although, as I say, it has no necessary bearing on the sense, those who are more interested in whether such things have any possible relevance may find my master’s thesis diverting: http://harbeck.ca/James/Harbeck_James_C_2016_MA.pdf (PDF, 10 MB).
Just a bit more about what I’m doing with the last two paragraphs, from my “About WTNs” page:
Word tasting notes are not simple accounts of facts; they deliberately go further in aestheticizing the verbal object, just as the object of wine tasting is not simply to evaluate and describe but to find as much as one can in the wine. So the various visual features and echoes noted in the word tasting notes may or may not have any effect on the day-to-day use of the word; the fact that similarity of sound does not always have much if any effect can be illustrated by how often a person will be surprised by the connection made by a pun.
Thus, I am not perpetuating the phonaesthetic fallacy; I do not believe that sounds have intrinsic aesthetic values – aesthetic value is not intrinsic in anything; it is always a function of the interaction between the stimulus and the perceiver – and I do not believe that sounds necessarily determine the meanings of words. Most of the time they don’t have an important influence (unless and until we let them), but we can see that in some words phonaesthetics clearly do have some influence on meaning and usage – notably words formed on the basis of onomatopoeia and sound symbolism, but also words that have established phonaesthemes in them, for instance the /gl/ and /sn/ onsets.
The point of word tasting, then, is to stimulate aesthetic appreciation of words, to help people have fun with them and look at them from various angles and in light of various aspects and influences, in search of both information and delight. If you taste a word, you will undoubtedly get different things from it than I do. But I certainly hope that what I get from it will stimulate to get more than you might have.