By "standard-type valuation", I will first assume you mean under a GBM process. All you need is the joint density of (ST,MT): the terminal value and the minimum of the process over (0,T).
That density is known in closed form for drifting BM and hence GBM. (Googling shows it's in Mark Joshi's 'Concepts.." book, for example)
Then, the option value is simply the (discounted, double) integral of that density times the payoff function that you gave in your post. (Do the integral numerically and you're done -- if a GBM value is all you really wanted).
Of course, maybe you'll want to go on from there to a more market-consistent model. It would be nice to have the stand-alone ST density agree with the market's RN density which you can infer from Breeden-Litzenberger, for example. Can you infer a model-independent, market-consistent joint (ST,MT) density? Hard to say -- likely depends on what other types of options are liquid and well-quoted. You may need to calibrate various more complicated models to the vanilla (and other liquid barrier) options, and then do Monte Carlos to price the thing.