Judge Bonham denied a motion to allow members of the jury to go home a

Let's first go through a bit of conceptual theory regarding the use of verb+ing words.

1. Just because an 'ing' is attached to a verb, it doesn't become a gerund. In fact, there are three forms of ing that one will come across in formal writing. The first is the present participle, (with or without a comma before). Such an "ing" word is always a modifier and never a gerund.

Here in this example, the use of allowing without a comma before modifies the motion and with a comma before 'ing' word modifies Judge Botham and his denial. In both cases, however, the present participle is a modifier.

2. To become a gerund the 'ing' word must act as a noun and will mostly be either the subject or object of a clause. If it is a subject, it will most probably start a clause or at least be a part of a noun phrase that will in sum be the subject. What is more important is that such a word or phrase will be immediately followed by a verb to corroborate that what you have before is indeed a subject. In some cases, there will be possessive adjectives before 'the' ing word or an article acting as an adjective. Since possessives and adjectives modify nouns, we can be sure that the following 'ing' word must be a gerund.
Example
Being addicted to drinks has ruined many a person. 'Being addicted' is a gerund and a noun phrase followed by its verb 'has ruined'.
Exercising is a good discipline if done regularly -- Here exercising is a gerund with its verb 'is'
An 'ing' gerund is a legal application in GMAT.
3. The third use of verb+ing is to use it as a part of a passive voice verb, considered legal in GMAT.
Example

He looked peeved when the inquiry committee was questioning him.

With this bit of theory in mind, one can find that the use of present participles is out of place in the context and hence wrong.

Now on to the official choices
Judge Bonham denied a motion to allow members of the jury to go home at the end of each day instead of to confine them to a hotel.

(A) to allow members of the jury to go home at the end of each day instead of to confine them to
(B) that would have allowed members of the jury to go home at the end of each day instead of confined to
(C) under which members of the jury are allowed to go home at the end of each day instead of confining them in
(D) that would allow members of the jury to go home at the end of each day rather than confinement in
(E) to allow members of the jury to go home at the end of each day rather than be confined to

First of all, let's ruthlessly reject choices A, B, and C for using the unidiomatic 'instead of'.

Between D and E, the infinitive 'to allow' pointedly indicates that the motion, yet to be approved then, had a definite purpose inherent in it, while in D 'that would allow' is more indicative than intentional. Therefore, E is the best.