喜んで is just the 〜て form of 喜ぶ. And as you (might) know, one main usage of the 〜て form is to connect multiple clauses as "and":
- 喜びます → I am/will be glad.
- あなたのお手伝いをします → I will help you.
- 喜んであなたのお手伝いをします。 → I am pleased and will help you → I will be pleased to help you.
You could also think of this 〜て as the "by means of 〜" form (for which 〜で is used on nouns).
- 電車でゴー！ → Let's go by train
- 魚を焼いて食べる → Grill a fish and eat it ("Eat a fish by grilling it").
So you could think of this sentence as
I will help you "by means of" being happy.
The form here is simply the 〜て form. There is nothing going on concerning the passive voice (受身) or imperative (命令形). The passive form of 喜ぶ is 喜ばれる. The imperative form is 喜べ, and while the 〜て form can function as a "light" imperative (as 〜てください, omitting the ください), that is not the case here. If the sentence were slightly changed to say something like the following, then you'd have a "light" imperative.
- 喜んでね、あなたのお手伝いをします。 → Cheer up, I'll help you out.
But then the context of the sentence is somewhat different; you're comforting the other person who almost seems bummed out about some large task and needs help to get it done, whereas in the original sentence the person's feelings about needing help feel more neutral.
Remember that the present and future tenses in Japanese are the same, and although context can often help figure out which it should be, there are times when it's ambiguous. In this case it's clearly the future because it would be strange to say to someone, "I'm gladly helping you (now)!"