The question didn't carry the [tag:british-english] when I replied, so be aware that this answer explains the American English style.

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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<sup>1</sup>, 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.
                        ^   ^        ^
                        +---+--------+

In this example it modifies the predicate or object, not the subject.  And that will be the natural line of thought when hearing a sentence in which the modifier is placed after the verb.

Also correct:

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

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

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<sup>1</sup> Note that I am not saying the constructs offered by the exercise are outright incorrect from a grammatical perspective.  But they slow down comprehension<sup>2</sup>, compared to sentences starting with the unambiguous clause "Both of them".

<sup>2</sup> For a discussion of ambiguity and its effect on comprehension time and effort, one should consult the cognitive science literature, such as [Gerry Altman "Ambiguity in sentence processing"](http://www.cell.com/trends/cognitive-sciences/abstract/S1364-6613(98)01153-X)

Language learners would do well to learn the practical aspects of clear communication in addition to the borderline cases permissible according to grammar yet unclear.