DBeaver, How to declare variables and use them?

You have to enable variable processing in the “SQL Processing” settings of DBeaver -> Window -> Preferences -> Database -> Editors -> SQL Editor -> SQL Processing. There is a block on Parameters with settings you can change. See the Dynamic Parameter binding section on the wiki.

You should then be able to do:

@set date="2019-10-09"

SELECT ${date}::DATE, ${date}::TIMESTAMP WITHOUT TIME ZONE

which produces:

| date       | timestamp           |
|------------|---------------------|
| 2019-10-09 | 2019-10-09 00:00:00 |