In the first high-level language (FORTRAN), lines of text would by default be treated as individual statements, except that a card which contained something other than an asterisk or C in column 1 and something other than a blank or zero in column 7 would be treated as a continuation of the previous line. In most subsequent languages which had a concept of "statements", they were either indicated by explicit ending delimiters (typically semicolons), by continuation markers placed at the end of the line (e.g. -_ in VB.NET), or by a grammar that would leave things unresolved at the end of a line (JavaScript).
If a language will be interactively processed by a REPL that receives lines of inputs, it's useful to have the text of a line indicate whether the contents of the line should be processed without awaiting further input. In all scenarios involving static source files, however, I would think the essence of the very first approach that was ever used (using something at the start of the line after the incomplete one, rather than at the end of the incomplete line itself) would be superior. A tiny amount of extra work might be needed to perform "lookahead" before processing each line, but if such a burden was tolerable for the first FORTRAN compilers it should be no less so today.
Have languages other than FORTRAN indicated continuations using a prefix on the line containing extra content, rather than a suffix on the incomplete line?