Would you please tell me if the following sentences are gerund or present continuous?

Jean stays with us quite often. The children love having her here.

Jean's with us at the moment. The children are loving having her here.


having is a gerund, in both examples.

The structure is They love something.

eg They love chocolate.

eg They love swimming.

eg they love having her here.


Only your second example is in the continuous aspect:

[1] The children love [having her here].

[2] The children are loving having her here.

In [1] having is a gerund-participle verb functioning as head of the bracketed clause.

In [2] "are" is the continuous auxiliary complemented by the gerund-participle "loving". Together, "are+"loving" form the the continuous aspect.