Everton winger Aaron Lennon is currently receiving treatment for a stress-related illness after being detained under the Mental Health Act.
"Players have always been under pressure from fans and media but now it is massive," said Bilic.
"With such a black and white media, one day you are king, the next a zero."
He added: "Everyone involved in football should do more, especially with young players. We are forgetting they are like kids with ability, but a bit more fame and money. It is not easy to cope with pressure."
The Professional Footballers' Association, the players' union, said on Wednesday that more players than ever require help for mental health problems, with 62 current and 98 former players seeking support last season.
Everton manager Ronald Koeman said the club was endeavouring to "support" Lennon's "private situation".
"We try always with every individual player to help him improve as a football player and if he feels some problems in mental or other issues we have enough people working to help players with these kind of problems."
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The PFA set up a welfare department in 2012 which offers a range of support options for players, including a 24-hour phoneline and access to a psychiatrist.
West Ham boss Slaven Bilic wants more support for players facing mental health problems and says pressure in the game has never been so intense.