两种可能性：offering candy看成动名词。

1):Be 谓语动词， careful of strangers  offering candy为形容词词组作表语。在形容词词组中，中心词是careful, of介短短语作形容词补足语; strangers offering candy是动名词复合结构,作of的介词宾语(或介词补足语)。句子意思就是小心陌生人给你糖果(这个行为),提醒孩子们要注意“某事”

2):Offering candy看成现在分词作后置定语。 Be careful of strangers offering candy= Be careful of strangers who offer candy就是小心给你们糖果的陌生人.  提醒孩子们要注意“某人”

我认为问题都不大。

本人观点：strangers offering candy 英语思维这里是strangers who are offering candy. 当心那些在给糖的陌生人。当心的是那些人，如果是动名词，那是要当心有陌生人发糖这种做法。offering candy 应该是分词作后置定语才符合英语思维。如果语法书解释为动名词，这种解释不符合现代英语中的思维逻辑。在我印象里，国外语法课讲gerund 一般仅用于作主语，宾语和定语。

补充：刚才我在国外WR论坛对这个问题做了提问，外国人所有回复 都认为是现在分词，不是动名词。

https://forum.wordreference.com/threads/be-careful-of-strangers-offering-candy.3833993/#post-19588862

这是我的提问：
Hi all，

In this sentence: Be careful of strangers offering candy in the street! Could anyone please explain whether the word "offering" here is a present participle or gerund? I think" strangers offering candy" in this context should mean strangers who are offering candy. Therefore offering is a present participle modifying strangers. Am I right?

Thanks in advance!

1. 英国人高级会员的回复：I think you're right.  

2. 英国人回复：If it were a gerund you would have three nouns together which would be unusual I would think, maybe impossible given the importance of word order in English.

3.英国人回复："Offering" is complemented by its object "candy", so it's a (non-finite) verb here. "Offering candy in the street" is a participial clause that modifies "strangers".