Assuming you know your keys, circle of 5ths and 4ths etc. You could start by looking at the simplest key that contains the accidentals in you phrase (not the only approach but hopefully a logical starting point).
The sequence of flats is (B, E, A, D, G, C, F) so the Key of Eb Major is possible.
Then ask yourself if the Melody seems for favor notes from the Eb chord (this is assuming that the composer intended to be on the I in the phrase, and this type of analysis requires assuming, then testing your assumptions, then changing your mind until your assumptions make musical sense. In time you get better and making good assumptions).
In my opinion the melody does not really look like an Eb centric melody. It looks more Ab centric. But this does not really mean that you are in Ab. You could assume the key is Ab with (B, E, A, D) flats. This does not contradict the information given. Then the natural chords may be the I (Ab), unless you are looking harmonize each note as in classical Homophony. But it could be that this phrase is in the key of Eb and centered around the IV chord (which would be the Ab).
Without any additional context you could go either way. Was this one phrase from a bigger piece or an original composition that you are trying to do something with? That would help us help you.
As for picking the correct chords to play with the phrase, that could go in a couple directions. If you want Homophony you need to identify a key choice then pick chords that contain the note in question and move gracefully form the chord before and to the chord after. There is a whole science behind this with rules and regulations (too much for a single post). If you just want to vamp behind the phrase and not be obtuse you can look for chords that contain the "featured" notes. In this case one choice is to drone on Ab. The notes that are not exactly in that triad (Ab, C, Eb) may still sound nice (being 9ths and Major 7ths), or act as passing tones. The Bb, for example could be harmonized by the V chord to create some movement.
I hope some of this helps but in fact you are asking for a lot given so little.