As the comments say, both alternatives are worse than "they both have".

However, in context there is a clear advantage to one of them for reasons of parallel construction.

The assignment appears to have precircled "they are both (tall)", thus the complete sentence would be

    They are  both tall and
     ||        ||
    they have both got brown eyes.

However, the pre-chosen phrase is also bad style, it should be

    (They) Both are tall and both have brown eyes.     (Good)

Contrast this with correct use of both following the verb

    My grandmother is both old and wise.
                        ^   ^        ^
                        +---+--------+

When placed after the verb, it modifies the predicate or object, not the subject.

Also correct:

    They are both tall and brown-eyed.
               ^   ^           ^
               +---+-----------+

but this is usable whether "they" are two or many, because "both" isn't linked with "they".

The other choice is similar:

    They both are tall and have brown eyes.
           ^   (-+-)         (----+----)
           |     |                |
           +-----+----------------+

"Both" still doesn't modify the subject.  If you wanted it to modify the subject, then

    Both of them are tall and have brown eyes.         (Good)
      ^  /\   ^
      +--  ---+

At best the preselected word order is ambiguous and confusing.