Any recommendations for day tours with Santiago Chile as base?

You definitely want to visit Valparaiso.

We went on a couple wine tours arranged through the hotel; Viña Undurraga and Viña Concha y Toro. We liked the Undurraga tour much better. While it wasn’t a tour, we took the local bus over to Valparaiso and spent the better part of a day there. It is a gritty port town and back in 2014 one needed to be aware of one’s surroundings since there was a lot of petty crime. Also, when we were there the funiculars were not functioning so we didn’t get to ride one of them in that city. Overall we enjoyed both Santiago and Valparaiso.

In March 2015 my husband and I spent two weeks in Chile, which included 3 days on Easter Island and 4 days at Tierra del Fuego, taking a three day cruise on a small boat (150 passengers) exploring the small islands at the "bottom" of South America (e.g. Isla de los Penguinos). During our wonderful stay in Santiago, we took a day trip (10-11 hours) to the Colchagua Valley for wine tasting and lunch. It was with Uncorked Tours, and one of the highlights of our Santiago stay. It was semi-private (there were seven people), they picked us up at our hotel and returned us there. We visited 3 vineyards, all of which were VERY generous with their tastings. The lunch we had at one of the vineyards was like a deluxe dinner at a fine restaurant, and also included wine. The guide was friendly, knowledgeable and helpful, and we learned a lot about Chile's agricultural history and wine culture. It was one of the more expensive tours, but worth every penny. The Colchagua Valley was about 2-2 1/2 hours from Santiago, but time flew on the road thanks to our guide. It was $255.00 per person for this tour, high-priced for us, but we are so glad we went!
I would book with them any time. I don't know what their prices are today, but they did a great job. (Just fyi, we don't usually do tours, yet this was money well-spent.)