Safari does not support saving the credentials for basic authentication. That's why you're not asked to store it.
There are basically three work-arounds you can use:
The first is simply to create a bookmark that contains your credentials. Note that this should only be done if the credentials aren't "important" - like for example if this is one of those commercial web sites that allow anyone in that knows that this year's password is "2019" or something like that. Then you can create a bookmark for the URL like this:
When you open that bookmark, you'll not be asked for credentials, but will be logged in automatically.
If you require an ordinary amount of security regarding the saved credentials, you can instead use a different work around. On your device, access Settings and then Passwords & Accounts and then App & Website Passwords. At the top is a + icon that you can click to add your web site and the username and password.
Contrary to form based password authentication, the information you enter there will not be automatically filled out. However when you get the username/password popup, you have the option of clicking the little "key" icon to select the stored credentials.
The third possible work-around is to use a Mac logged in to the same iCloud account with Keychain enabled. Access the web site using Safari on the Mac, and you'll be able to store the credentials.
On the iOS device the credentials will not be automatically filled out, but you will be able to press the "lock" icon in the popup to allow choosing the credentials you've stored on the Mac.