Helium balloons cause hundreds of train delays every year
The publicly-owned company, which is responsible for Britain's rail infrastructure, recorded 619 balloon-related incidents across Britain in the past 12 months.
Many of the episodes include balloons getting tangled around high-voltage overhead wires, causing delays as the electricity supply must be turned off while they are removed.
James Dean, chief operating officer for Network Rail's London North Western route, said: "If you're on a railway station platform with a foil balloon filled with helium on a string and it comes in contact with the overhead wires carrying 25,000 volts, that could cause huge injury or death.
"Ideally, we'd ask people not to bring balloons into our stations at all.
"Alternatively, carry them in bags so the risk of them floating upwards is minimised."
As well as the safety risks and the delays to passengers, the annual cost of the problem to taxpayers is estimated to be around £1 million due to compensation pay outs.
Hundreds of passengers were delayed earlier this week when a float-away helium balloon got wrapped around overhead wires at Smethwick Rolfe Street station in the West Midlands.
The cost of delays from this single incident was £5,000.
George Oustayiannis, chairman of trade association the National Association of Balloon Artists and Suppliers, said: "Balloons bring fun and colour and a sense of celebration to any event, but please dispose of them responsibly, and never release balloons into the atmosphere.
"Respect the environment and prevent unnecessary danger and delays."