How do I address an envelope to a married couple, who are respectively Professor and Dr?

Solution 1:

In British custom, one would write on the envelope

Prof J R Ewing and Dr E M Ewing

Note that the surname is repeated, because the wife is notable in her own right and not merely as "Mrs J R Ewing". This is particularly the case if the wife is still using her maiden name:

Prof J R Ewing and Dr E M Crump

If you wanted to be particularly formal, then post-nominal letters can be added, again separately:

Prof J R Ewing OM and Dr E M Ewing FRCP
Prof J R Ewing OM and Dr E M Crump FRCP

In formal communications, the husband is always addressed first, even if he's just a plain Mr. However in this case it may be preferable to use the slightly more formal Esq, particularly if the wife has post-nominal letters; again, because the wife is notable in her own right, she's accorded everything going. If you choose to use Mr instead of Esq then make the wife match:

J R Ewing Esq and Dr E M Ewing FRCP
Mr J R Ewing and Dr E M Ewing
Vice Admiral Sir T J H Laurence KCVO CB CSM ADC(P) & HRH Princess Anne, The Princess Royal KG KT GCVO GCStJ QSO GCL

None of this precludes an ordinary "Mr & Mrs J R Ewing", or (if he has a qualification and she doesn't) "Dr & Mrs J R Ewing". The point is that if the couple has to be dealt with separately because the wife has stuff to be listed in her own right, then they are written separately in order that that can be made clear. It would never be right to write to "Mr & Dr J R Ewing", and even "Mr J R & Dr E M Ewing" is not strictly correct, although often seen and certainly more acceptable than the former.

Debrett's on joint forms of address
Debrett's on post-nominal letters