Does God exist ?

How do we know God exists ?

To Exist

To answer this question,  let us first us first understand what does it mean when we say something 'exists'.If we say, we know an object A exists, if it can be verified by some person, by means of some sensory perception.This definition of of an object 'existing', requires another object to exist (the person verifying it).  Thus to define the word to 'exist', we are required to make use of another object which 'exists'.This results in an infinite regress  or circular reasoning and hence using this approach we will never arrive at what to 'exist' means.

• Let us elaborate this,  to say object A (sentient or insentient) is existing would require another sentient object B which would perceive object A. Thus we are defining object A  as existing if another object B is existing and which is sentient and perceives A. Here to define 'existing' we are doing it using 'existing' and 'sentient'. Now for object B to exist, it would require another object C which perceives it and so on which would result in infinite regress. 
• The other way would be circular, where A is perceived by B and B is perceived by A, i.e., Object A's existence would require object B's existence and object B's existence would presuppose Object A's existence, hence we are back to where we started. 

In both these approaches, we have not been able to define 'existence' in terms of something else.
 We are thus forced to conclude that 'existence' is fundamental, it can not be broken down or defined by something else.

Real and Unreal

Consider two balls A and B, both with the same set of attributes. For example both are red, both have the same size, weight, texture etc. But ball A exists and ball B does not exist. That is A is real and B is not real or imaginary. What is it exactly that makes A real and B unreal, is the fundamental element of 'existence' present in A and not in B.

God 

Now let us take the first part of the question. Who/What is God ?
Below is how God/Parabrahma is defined in the Upanishads.

In बृहदारण्यकोपनिषत् we find in 2.1.20
स यथोर्णभिस्तन्तुनोच्चरेद्
 यथाऽग्नेः क्षुद्रा विष्फुलिङ्गा व्युच्चरन्त्य्
 एवमेवास्मादात्मनः सर्वे प्राणाः सर्वे लोकाः सर्वे देवाः सर्वाणि
भूतानि व्युच्चरन्ति ।
तस्योपनिषत्सत्यस्य सत्यमिति
प्राणा वै सत्यं
तेषामेष सत्यम् ॥ २.१.२० ॥

Translation: As a spider moves about on its own net; as innumerable tiny sparks come forth from the blazing fire, even so from the Lord emanate all the jiva souls with particular characteristics manifested owing to connection with limited adjuncts: (i) the gods here mukta jivas (nitya muktas) and all the animate and inanimate objects.
The mystery of Upanishad thereof is that God is the Real of the reals. The jiva souls are also real and of them God is the reality.

Here it is said तस्य उपनिषद् सत्यस्य सत्यम् इति, i.e., the essence of the Upanishat says, that सत्यस्य (of the existing ), God is the सत्यम् (existence).  [भू  is the first verb in Panini's dhatu patha, which means existence सत्तायाम् . Truth, Reality and existence are synonymous. ].

Hence we see that Parabrahma/God is defined as that fundamental element in objects, which makes any object real.  Like in Bhagavad Gita Parabrahma Sri Krsna says  मयि सर्वं इदं प्रोतं सूत्रे मणि गण इव , i.e., Everything rests upon Me(God), as pearls are strung on a thread. Thus we see by the Vedic tradition defines God as the very essence of existence.

Now, that we have discussed what is existence and what is God (By the Vedic tradition), let us try to answer the question, 'Does God exist ?" . Now as 'exist', can not be further broken down to be defined by any simpler concept. And as God is nothing but the essence of all existence or existence  itself.
Thus the question 'Does God exist ? ' translates to 'Does existence exist ?'. The answer has to be a 'Yes!'