The rail operator warned of "multiple line closures, flooding, trees on lines, and loss of power" across Yorkshire and the North West.
"Our advice is not to attempt to travel as we are unable to provide alternative transport," it stated on Monday morning on Twitter.
It also announced trains would be unable to stop at Rotherham Central for the rest of the day, sharing a shocking photo showing flooding between the station's platforms.
Meanwhile, Leeds City Station warned of route closures as it urged travellers to check their journey could still go ahead before setting off.
"Be prepared that you may not be able to travel," it added.
British Transport Police South Yorkshire said: "#Rotherham station resembles the canals of Venice!
"Wind and rain is having an impact on the network this morning with delays and cancellations in place, the advice is to check before you travel."
Further images show other sections of the track completely flooded - which more rain feared as Storm Franklin arrives.
Matt Rice, route director for Network Rail, said: “As we have seen across much of the network this week, severe weather brings severe challenges for the railway.
"My team have spent the weekend battling the elements but such heavy downpours brought by Storm Franklin have caused multiple sites to flood, which means trains are not able to serve some places for the start of service this morning.
“We would like to reassure passengers that we’ll be doing all that we can to get them moving again and we thank all those impacted for their patience."
As of Monday morning, the Environment Agency had issued 108 flood warnings around Yorkshire.You can see the full list of service disruptions across the Northern network here.