Banana leaves weep at night. No problem, it is natural.
Be careful, the water that comes out stains things. I had one that dropped water on a glass top table and I could never get the stain off.
I don't think it can get too much sunlight indoors. Water it when it is dryish (not dry- they like moisture, but don't let it stay wet). Weekly sounds ok.

Veadie,

I Live In Dallas Texas and have a Dwarf Cavendish too. My Banana tree will sometimes do this, it is not water, but latex sap. It can make your skin very irritated if you touch it. It is nothing to worry about. All banana trees do it in times of stress. :-)

You should have learned in high school biology that that is called "gutation" which you can now Google to your heart's content instead of listening to jackasses, such as I, pontificate.

I have always taken gutation as a sign of a healthy root system because gutation cannot occur without a healthy root system.

Guttation :).

Good to know it's not latex sap.... I tried a few drops last year and it tasted like water. I was thinking of making "Banana Water"