We have rented an apartment on Rue Linne near SW border of Jardin des Plantes.
What is the best train station to get us close to Giverny
You do not have a choice: trains to Vernon-Giverny leave from Gare Saint Lazare, which you can access by metro line 7 to Chaussée-d'Antin-Lafayette followed by a 10-min walk, or get off one stop before at Opéra and take line 3 to Saint-Lazare.
Edit: also, buses 21 and 27 go direct to gare Saint Lazare from near the bottom (north end) of Avenue des Gobelins. It is a nice ride but travel time is less reliable than by metro, so not ideal if you have a train to catch!
Thank you