The music you have shown would likely be played as a bass line in a fingerpicking style. For this, I would probably just use the thumb with all down strokes. This line would likely be combined with one or more other parts to be played by the other fingers, but the bass line is important to the feel of the boogie.
Here is an example 12 bar blues in tablature using the boogie bass line from the question. It is important to maintain some independence between the bass line and the upper voices. The fingerings here are not complicated, but it may take some practice to be able to execute them smoothly.
$E 5 $E.5.$G.6.$B.8 $A 4 $A.4.$G.6.$B.8 $A 7 $A.7.$G.6.$B.8 $D 4 $D.4.$G.6.$B.8 | $D 7 $D.7.$G.6.$B.8 $D 4 $D.4.$G.6.$B.8 $A 7 $A.7.$G.6.$B.8 $A 4 $A.4.$G.6.$B.8 |
$E 5 $E.5.$G.6.$B.8 $A 4 $A.4.$G.6.$B.8 $A 7 $A.7.$G.6.$B.8 $D 4 $D.4.$G.6.$B.8 | $D 5 $D.5.$G.6.$B.8 $D 4 $D.4.$G.6.$B.8 $A 7 $A.7.$G.6.$B.8 $A 4 $A.4.$G.6.$B.8 |
$A 5 $A.5.$B.7.$e.8 $D 4 $D.4.$B.7.$e.8 $D 7 $D.7.$B.7.$e.8 $G 4 $G.4.$B.7.$e.8 | $G 5 $G.5.$B.7.$e.8 $G 4 $G.4.$B.7.$e.8 $D 7 $D.7.$B.7.$e.8 $D 4 $D.4.$B.7.$e.8 |
$E 5 $E.5.$G.6.$B.8 $A 4 $A.4.$G.6.$B.8 $A 7 $A.7.$G.6.$B.8 $D 4 $D.4.$G.6.$B.8 | $D 7 $D.7.$G.6.$B.8 $D 4 $D.4.$G.6.$B.8 $A 7 $A.4.$G.6.$B.8 $A 5 $A.6.$G.6.$B.8 |
$A 7 $A.7.$G.7.$B.9 $D 6 $D.6.$G.7.$B.9 $D 9 $D.9.$G.7.$B.9 $D 6 $D.6.$G.7.$B.9 | $A 5 $A.5.$G.5.$B.7 $D 4 $D.4.$G.5.$B.7 $D 7 $D.7.$G.5.$B.7 $D 4 $D.4.$G.5.$B.7 |
$E 5 $E.5.$G.6.$B.8 $A 4 $A.4.$G.6.$B.8 $A 5 $A.5.$G.6.$B.8 $A 6 $A.6.$G.6.$B.8 | $A 7 $A.7.$G.7.$B.9 $A 9 $A.9.$G.7.$B.9 $D 5 $D.5.$G.7.$B.9 $D 6 $D.6.$G.7.$B.9 | $E.5.$G.6.$B.7 ||
It is also worth noting that when tempos speed up (or even at slower tempos), it is not uncommon to omit some of the bass notes when the upper voices help to establish the rhythmic feel:
$E 5 $G.6.$B.8 $A 4 $G.6.$B.8 $A 7 $G.6.$B.8 $D 4 $G.6.$B.8 | $D 7 $G.6.$B.8 $D 4 $G.6.$B.8 $A 7 $G.6.$B.8 $A 4 $G.6.$B.8 |
Concerning these Chord Voicings
The OP has asked about why I played these chords starting at the fifth fret, noting that this could also be played with open strings. This is true, at least in part. It is good to play things in different positions and with different fingerings, and there is no reason that the above must be played exactly the way I wrote it.
To start, note that the chords used above are shell voicings, which are chords composed of the root, third, and seventh of a chord. These notes contain the essential skeleton of a seventh chord, and are very handy to have in your hip pocket. There is plenty of interesting stuff to say about shell voicings, but that would take us too far afield, I fear. Here are some shell voicing chord diagrams (apologies for the chord diagram rendering-- I couldn't get jTab to render more than two diagrams side-by-side):
%5/1.X/X.X/X.6/2.8/4.X/X[A7]
%X/X.5/1.X/X.X/X.7/3.8/4[D7]
%X/X.7/1.X/X.7/2.9/4.X/X[E7]
Note that the D7 here is voiced with the F♯ and C (the 3rd and the 7th) on the top two strings. If this chord is moved to use an open D, those top notes won't be reachable, so either you would have to move up to the fifth fret after playing the A7, or you could voice the chord differently.
If you are paying attention, you will notice that there are two voicings for a dominant 7th chord with the root on the fifth string shown above. Here are three such possibilities for C7:
%X/X.3/1.X/X.X/X.5/3.6/4[C7]
%X/X.3/1.X/X.3/2.5/4.X/X[C7]
%X/X.3/2.2/1.3/3.X/X.X/X[C7]
And here are four possibilities for a G7 rooted on the sixth string (the first one requires a stretch, and may be easier higher on the fingerboard):
%3/1.X/X.X/X.X/X.6/3.7/4[G7]
%3/1.X/X.X/X.4/2.6/4.X/X[G7]
%3/1.X/X.3/2.4/3.X/X.X/X[G7]
%3/2.2/1.3/3.X/X.X/X.X/X[G7]
So, I voiced the D7 starting in the fifth bar the way I did because the bass line ends up on the G string, and I didn't want it to interfere with the top notes of the chord. I couldn't do this with an open D in the bass; since I needed to put the D7 at the fifth fret, I also put the A7 at the fifth fret to reduce movement around the fretboard.
You may want to try different ways to play this, and note that there isn't really a "right" way. You might not like the way it sounds, and prefer the sound of some open strings, or might prefer the sounds of some different voicings. You might like to change some of the chords to 6th chords (the final chord is an A6). You might find this too difficult to finger in the written positions. Any of these would be a good reason to find new ways to play this.
Experiment. Hopefully the above chord diagrams will help you to see other ways to play with the original bass line.