When you purchase wire basket with your beach cruiser here are the parts you should receive.

U shaped wire brace
4 bolts and 4 nuts
Flat bracket for securing the top part of basket
Shaped bracket for securing the wire brace

Bracket with a hole in it to attach to the handlebar
(can be already installed on the bike or shipped separately together with basket).

First attach the U shaped wire brace using the shaped bracket to the basket.

Use two nuts and bolts and tighten them up.

Loosen the nuts on each side of the front wheel and remove them.
Attach wire brace and tighten the nuts again.

Now you have to secure the top part of basket to the handlebar bracket
(either already installed you have to install it by removing the handlebars first).

It is all done.
Congratulations!