Generally, on the harpsichord you hold down notes when it will increase the amount of resonance in the tone of the instrument. At risk of stating the obvious, the harpsichord does not have anything equivalent to the piano's sustain pedal.
So it would certainly by idiomatic to hold down all the notes in the arpeggios as they are played. You might consider releasing the held notes when you play the dotted-8th top G, so you can make a clear articulation with some silence between the two top G's, rather than have the previous chord still sounding. (In other words, try playing the dotted-8th G as an 8th-note followed by a 16th rest.)
It would also be idiomatic to arpeggiate the half-note chord as well. Play the top and bottom notes (E and G) together on the beat, and let the other notes follow, quickly (probably faster than the written-out arpeggio).
From bar 3 onwards, you could hold down the two left-hand notes at the start of each beat as well - try releasing them on the top note of the right-hand (i.e. for the first beat, hold them for the B and C and release them when you play the D.
Finally, don't take the written rhythms too literally - it's OK to use a lot of rubato, so long as the overall "shape" of the musical phrases makes sense. For example, the very first note could be a lot longer than the written 16th-note - imagine this is an improvisation, where you play that E and then "pause while you think what to play next" before playing the rest of the arpeggiated chord.
The most significant difference between piano and harpsichord in situations like this is not that the harpsichord notes don't sustain as long as piano nots, but the fact that the harpsichord has a strong aggressive attack on every note, however short it is, which is impossible to imitate on the piano.