What is it permissible for a man to do with his wife during the day in Ramadaan?

Yes, that is permissible. In fact it is permissible for a man
to be intimate with his wife when he is fasting, so long as no intercourse
or ejaculation takes place. 

Al-Bukhaari (1927) and Muslim (1106) narrated that ‘Aa’ishah
(may Allaah be pleased with her) said: “The Prophet (peace and
blessings of Allaah be upon him) used to kiss and be intimate when fasting,
and he was the most able of you to control his desire.” 

Al-Sindi said: 

The word yubaashir (translated here as “be intimate”)
means touching the wife’s skin and her touching him, such as putting his
cheek against hers and so on. 

What is meant here is touching the skin, not intercourse. 

Shaykh Ibn ‘Uthaymeen (may Allaah have mercy on him) was
asked: 

What is it permissible for a man who is fasting to do with
his wife who is fasting?

 He replied: 

A man who is observing an obligatory fast is not permitted to
do anything with his wife that will lead to ejaculation. People vary in how
quickly they reach climax; some of them are slow and can control themselves
completely, as ‘Aa’ishah (may Allaah be pleased with her) said concerning
the Messenger of Allaah (peace and blessings of Allaah be upon him),
“he was the most able of you to control his desire.” Some of them cannot
control themselves and ejaculate quickly. Such a person should not be
intimate with his wife and touch her or kiss her etc when observing an
obligatory fast. If a person knows that he can control himself, then he can
kiss and hug his wife even when observing an obligatory fast, but he must
beware of intercourse because having intercourse (during the day) in
Ramadaan for one who is obliged to fast results in five things: 

1-    
It is a sin

2-    
The fast is invalidated

3-    
He should refrain from eating
etc for the rest of the day. Everyone who breaks his fast in Ramadaan with
no legitimate shar’i excuse has to refrain from eating etc (for the rest of
the day) and also has to make up that day.

4-    
He has to make up that day
because he forfeited an obligatory act of worship, which he has to make up.

5-    
He has to offer an expiation
which is one of the most burdensome of expiations: freeing a slave; if he
cannot do that then he must fast for two consecutive months; if he cannot do
that then he must feed sixty poor persons. 

If it is an obligatory fast that was being observed at a time
other than Ramadaan – such as making up a day missed in Ramadaan, or
observing an expiatory fast etc, then breaking the fast results in two
things: sin and having to make it up.  But if it was a voluntary fast and he
has intercourse during it, then he does not have to do anything.