If you have changed your password, you should be okay. You can also add 2 Factor Authentication to your account for more security. If the attempts continue, you might want to consider a name change.
Note the restrictions in the article.
Apple ID Name Change
What to do after you change your Apple ID or password - Apple Support I suggest doing this before you change your Apple ID or password.
Apple ID -Two-factor authentication
Two Factor Authentification availability
Apple ID - Using app-specific passwords
Apple ID - Two Step Verification App Specific Passwords
Apple ID - Two Step Verifications FAQ
Apple won't send you an email/text telling you your account is disabled/restricted or will be disabled/restricted. You will find out the next time you try to log in somewhere.
Apple will always address you by your name or the name they have on file for you, not Dear Customer, Dear Client or by using your e-mail address. The e-mail will be from @apple.com or @iTunes.com. E-mail addresses can be spoofed. You can go to Mail/View/Message/Show all Headers to see more. Apple e-mails will never contain an attachment. Apple will never request personal information by email such as Social Security numbers, your Mother’s maiden name or full credit card numbers .
The only exception to the above I have noticed is if you order something from the Apple Store (apple.com), your receipt will be addressed to Dear Apple Customer. That is a receipt for a purchase you initiated.
Avoid phishing emails, fake ‘virus‘ alerts, phony support calls, and other scams
Identifying legitimate emails from the iTunes Store
Send the e-mail to Apple as an attachment to a new e-mail before deleting it. You can forward as an attachment by going to Mail/Message/Forward as attachment. Or control - click on the email and select Forward as attachment. You won’t receive a response.
reportphishing@apple.com