Thank you for including server version information - it makes answering your question easier.
You've not indicated which language(s) you are using.
Normally, though, you design a program to expect a certain schema (certain tables to be present), and then fail - preferably under control - if those tables are not present. Also, it is not clear whether you would get into problems because of repeated execution of the second INSERT statement. Nor is it clear when the NextRecordID table is updated - presumably, once the value has been used, it must be updated.
You should look at SERIAL (BIGSERIAL) and see whether that is appropriate for you.
You should also look at whether a SEQUENCE would be appropriate to use here - it certainly looks rather like it might be applicable.
As Adam Hughes points out, if you want to check whether the NextRecordID table is present in the database, you would look in the systables table. Be aware, though, that your search will need to be against an all lower-case name (nextrecordid).
Also, MODE ANSI databases complicate life - you have to worry about the table's owner (because there could be multiple tables called nextrecordid in a MODE ANSI database). Most likely, you don't have to worry about that - any more than you are likely to have to worry about delimited identifiers for table "someone"."NextRecordID" (which is a different table from someone.NextRecordID).