Milan is not the best “base” for 4-5 days of exploring the charms of Switzerland and Italy. It is a big, modern, bustling city more oriented toward business and finance than tourism. Hotels are generally expensive, and the city offers little in the way of antiquities or the “dolce vita” charm that most people seek in Italy. The Duomo is stunning, and the Last Supper is a world-class attraction, but both can be seen in an afternoon the day before one’s flight from Milan MXP or Linate.
If you want to show your family some of Switzerland and Italy, I suggest you take the train from Basel to Pontresina, an alpine village in the Engadine, 4.5 hours from Basel. With 2 nights there, most hotels will give their guests a pass for local transport, and you can ride a lift up to a restaurant with spectacular views, and a hike if you wish.
https://www.pontresina.ch/en/#
https://www.myswitzerland.com/en-us/destinations/pontresina/
On Day 3, you board the famed Bernina Express (or a regular regional train on the same tracks) for the scenic Bernina Pass journey over the Alps to the Italian border. The Bernina Express is a designated UNESCO World Heritage site. After 2 hours of breathtaking scenery, you arrive at Tirano on the border, walk across to the Italian side, and board an Italian train for the 1.5-hour journey to the lakeside village of Varenna on Lago di Como. This is the quintessential charming Italian resort-village, with hotels and other buildings spilling down the hillside to the lake. You can book rooms with balconies overlooking the lake at several hotels on the lakefront. We like Albergo Milano; others recommend Hotel du Lac or Villa Cipressi. They are all lovely.
https://lakecomotravel.com/varenna/
Spend 2 nights here if you can, then travel by train to Milan for the last night before your flight. It is only an hour by train from Varenna to Milano Centrale. Arrive in early afternoon and you will have ample time for the Duomo and/or Last Supper, and more. The Malpensa Express train will take you straight to the airport in just under an hour.