There is no direct connection from Euston Station to Epsom. However, you can take the Tube to Vauxhall then take the train to Epsom. Alternatively, you can travel from Euston Station to Epsom in around 35 min.
How to get from Euston Station to Epsom
From Euston Underground Station Tube to Vauxhall then train to Epsom. 41 min £9. From Euston Station taxi to Epsom. 35 min £82. From Euston Station drive to Epsom. 35 min £3.