It can still happen sometimes, but it's very rare that it crashes and even more rare that it corrupts as well. The good news, the chances of it corrupting your data silently was basically zero, then and now.
However back then MySQL seemed like it went out of it's way to corrupt your data. The only "bonus" is, it did it all silently, so nobody ever noticed until they went looking. With MariaDB(the successor) it's pretty rare that it silently corrupts your data these days.
I believe a lot of that can be tied back to the default switching from myisam to innodb. crashes frequently resulted in corrupt tables with myisam. innodb is very good at recovering.
Around early 00s, it was already common wisdom among the web devs that, when it comes to free RDBMS, if you want speed, you use MySQL; and if you want consistency and reliability, then Postgres is where it's at.
However back then MySQL seemed like it went out of it's way to corrupt your data. The only "bonus" is, it did it all silently, so nobody ever noticed until they went looking. With MariaDB(the successor) it's pretty rare that it silently corrupts your data these days.