Does Sneaky Deckhand enter ready if played to an open battlefield with The Exile's ability?

No — Sneaky Deckhand enters exhausted, because a battlefield that qualifies as "open" is by definition not one you control.

The Exile readies units played to battlefields you control, but an "open" battlefield is defined as unoccupied and uncontrolled (CR 170.11.c). Sneaky Deckhand's permission lets you play to a battlefield you don't control; playing there just applies Contested (CR 190.3.a.1) and stages a Non-Combat Showdown, so Control isn't established until that Showdown closes as a Conquer. Since The Exile's condition is checked as the unit is played/enters, it won't see you in control, and the default "units enter exhausted" (CR 359.2.c) applies.

If you instead play Sneaky Deckhand to a battlefield you already control (its ability is permissive, not required), The Exile would make it enter ready — that's the only way this works. This lines up with FAQ #12867, which covers Sai Scout/Sneaky Deckhand vs. the Riven legend; no separate official FAQ exists for this exact card pairing, so this is my interpretation based on CR 170.11.c and 190.3.a.1.